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NCERT Solutions for Class 11 Maths Chapter 15 Statistics are provided here. You have learnt the basic statistics like mean, mode and median known as "measures of the central tendency" in the previous Class NCERT syllabus. In this NCERT book chapter, you will learn important topics like measures of dispersion, mean deviation, standard deviation, quartile deviation, range and many more. In class 11 maths chapter 15 question answer, you will get questions related to all the above topics. These NCERT solutions are prepared by expert team at Careers360 keeping in my the latest syllabus of CBSE 2023-24, easy and simple language, step by step comprehensive coverage of the concepts.
This statistics class 11 solutions is important for the CBSE class 11 final examination as well as in the various competitive exams like JEE Main, BITSAT, etc . As statistics deal with the collected data for specific purposes and in the digital world, everything that we do comes in the form of data. This is a very useful technique to interpret, analyze the data, and make a conclusive decision based on the data. Class 11 maths chapter 15 NCERT solutions will build your fundamentals of statistics which will be useful to learn advanced statistical techniques. There are 27 questions in 3 exercises in the NCERT textbook. Interested students can find all NCERT solutions for class 11 at one place which will help to understand the concepts in an easy way.
Also Read :
Measure of Dispersion:
Dispersion measures the degree of variation in the values of a variable. It quantifies how scattered observations are around the central value in a distribution.
Range:
Range is the simplest measure of dispersion. It is defined as the difference between the largest and smallest observations in a distribution.
Range of distribution = Largest observation – Smallest observation.
Mean Deviation:
Mean Deviation for Ungrouped Data:
For n observations x1, x2, x3, ..., xn, the mean deviation about their mean x¯ is calculated as:
MD(\bar x) = (Σ |xᵢ - x¯|) / n
The mean deviation about its median M is calculated as:
MD(M) = (Σ |xᵢ - M|) / n
Mean Deviation for Discrete Frequency Distribution:
For a discrete frequency distribution with values xᵢ, frequencies fᵢ, mean x¯, and total frequency N, the mean deviation is calculated as:
MD(\bar x) = (Σ fᵢ |xᵢ - x¯|) / (Σ fᵢ) = (Σ fᵢ |xᵢ - x¯|) / N
Variance:
Variance is the average of the squared deviations from the mean x¯. If x_{₁}, x_{₂}, …, xₙ are n observations with mean x¯, the variance denoted by σ² is calculated as:
σ^{²} = (Σ(xᵢ - x¯)^{²}) / n
Standard Deviation:
Standard deviation, denoted as σ, is the square root of the variance σ^{²}. If σ^{²} is the variance, then the standard deviation is given by:
σ = √(σ^{²})
For a discrete frequency distribution with values xᵢ, frequencies fᵢ, mean x¯, and total frequency N, the standard deviation is calculated as:
σ = √((Σ fᵢ(xᵢ - x¯)^{²}) / N)
Coefficient of Variation:
The coefficient of variation (CV) is used to compare two or more frequency distributions. It is defined as:
Coefficient of Variation = (Standard Deviation / Mean) × 100
CV = (σ / x¯) × 100
Free download NCERT Solutions for Class 11 Maths Chapter 15 Statistics for CBSE Exam.
Class 11 maths chapter 15 question answer - Exercise: 15.1
Question:1 . Find the mean deviation about the mean for the data.
Answer:
Mean ( ) of the given data:
The respective absolute values of the deviations from mean, are
6, 3, 2, 1, 0, 2, 3, 7
Hence, the mean deviation about the mean is 3.
Question:2. Find the mean deviation about the mean for the data.
Answer:
Mean ( ) of the given data:
The respective absolute values of the deviations from mean, are
12, 20, 2, 10, 8, 5, 13, 4, 4, 6
Hence, the mean deviation about the mean is 8.4.
Question:3. Find the mean deviation about the median.
Answer:
Number of observations, n = 12, which is even.
Arranging the values in ascending order:
10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18.
Now, Median (M)
The respective absolute values of the deviations from median, are
3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5
Hence, the mean deviation about the median is 2.33.
Question:4. Find the mean deviation about the median.
Answer:
Number of observations, n = 10, which is even.
Arranging the values in ascending order:
36, 42, 45, 46, 46, 49, 51, 53, 60, 72
Now, Median (M)
The respective absolute values of the deviations from median, are
11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5
Hence, the mean deviation about the median is 7.
Question:5 Find the mean deviation about the mean.
Answer:
5 | 7 | 35 | 9 | 63 |
10 | 4 | 40 | 4 | 16 |
15 | 6 | 90 | 1 | 6 |
20 | 3 | 60 | 6 | 18 |
25 | 5 | 125 | 11 | 55 |
= 25 | = 350 | =158 |
Now, we calculate the absolute values of the deviations from mean, and
= 158
Hence, the mean deviation about the mean is 6.32
Question:6. Find the mean deviation about the mean.
Answer:
10 | 4 | 40 | 40 | 160 |
30 | 24 | 720 | 20 | 480 |
50 | 28 | 1400 | 0 | 0 |
70 | 16 | 1120 | 20 | 320 |
90 | 8 | 720 | 40 | 320 |
= 80 | = 4000 | =1280 |
Now, we calculate the absolute values of the deviations from mean, and
= 1280
Hence, the mean deviation about the mean is 16
Question:7. Find the mean deviation about the median.
Answer:
5 | 8 | 8 | 2 | 16 |
7 | 6 | 14 | 0 | 0 |
9 | 2 | 16 | 2 | 4 |
10 | 2 | 18 | 3 | 6 |
12 | 2 | 20 | 5 | 10 |
15 | 6 | 26 | 8 | 48 |
Now, N = 26 which is even.
Median is the mean of and observations.
Both these observations lie in the cumulative frequency 14, for which the corresponding observation is 7.
Therefore, Median, M
Now, we calculate the absolute values of the deviations from median, and
= 84
Hence, the mean deviation about the median is 3.23
Question:8 Find the mean deviation about the median.
Answer:
15 | 3 | 3 | 13.5 | 40.5 |
21 | 5 | 8 | 7.5 | 37.5 |
27 | 6 | 14 | 1.5 | 9 |
30 | 7 | 21 | 1.5 | 10.5 |
35 | 8 | 29 | 6.5 | 52 |
Now, N = 30, which is even.
Median is the mean of and observations.
Both these observations lie in the cumulative frequency 21, for which the corresponding observation is 30.
Therefore, Median, M
Now, we calculate the absolute values of the deviations from median, and
= 149.5
Hence, the mean deviation about the median is 5.1
Question:9. Find the mean deviation about the mean.
Answer:
Income per day | Number of Persons | Mid Points | |||
0 -100 | 4 | 50 | 200 | 308 | 1232 |
100 -200 | 8 | 150 | 1200 | 208 | 1664 |
200-300 | 9 | 250 | 2250 | 108 | 972 |
300-400 | 10 | 350 | 3500 | 8 | 80 |
400-500 | 7 | 450 | 3150 | 92 | 644 |
500-600 | 5 | 550 | 2750 | 192 | 960 |
600-700 | 4 | 650 | 2600 | 292 | 1168 |
700-800 | 3 | 750 | 2250 | 392 | 1176 |
=50 | =17900 | =7896 |
Now, we calculate the absolute values of the deviations from mean, and
= 7896
Hence, the mean deviation about the mean is 157.92
Question:10. Find the mean deviation about the mean.
Answer:
Height in cms | Number of Persons | Mid Points | |||
95 -105 | 9 | 100 | 900 | 25.3 | 227.7 |
105 -115 | 13 | 110 | 1430 | 15.3 | 198.9 |
115-125 | 26 | 120 | 3120 | 5.3 | 137.8 |
125-135 | 30 | 130 | 3900 | 4.7 | 141 |
135-145 | 12 | 140 | 1680 | 14.7 | 176.4 |
145-155 | 10 | 150 | 1500 | 24.7 | 247 |
=100 | =12530 | =1128.8 |
Now, we calculate the absolute values of the deviations from mean, and
= 1128.8
Hence, the mean deviation about the mean is 11.29
Question:11. Find the mean deviation about median for the following data :
Answer:
Marks | Number of Girls | Cumulative Frequency c.f. | Mid Points | ||
0-10 | 6 | 6 | 5 | 22.85 | 137.1 |
10-20 | 8 | 14 | 15 | 12.85 | 102.8 |
20-30 | 14 | 28 | 25 | 2.85 | 39.9 |
30-40 | 16 | 44 | 35 | 7.15 | 114.4 |
40-50 | 4 | 48 | 45 | 17.15 | 68.6 |
50-60 | 2 | 50 | 55 | 27.15 | 54.3 |
=517.1 |
Now, N = 50, which is even.
The class interval containing or item is 20-30. Therefore, 20-30 is the median class.
We know,
Median
Here, l = 20, C = 14, f = 14, h = 10 and N = 50
Therefore, Median
Now, we calculate the absolute values of the deviations from median, and
= 517.1
Hence, the mean deviation about the median is 10.34
Question:12 Calculate the mean deviation about median age for the age distribution of persons given below:
Age (in years) | ||||||||
[ Hint Convert the given data into continuous frequency distribution by subtracting from the lower limit and adding to the upper limit of each class interval]
Answer:
Age (in years) | Number | Cumulative Frequency c.f. | Mid Points | ||
15.5-20.5 | 5 | 5 | 18 | 20 | 100 |
20.5-25.5 | 6 | 11 | 23 | 15 | 90 |
25.5-30.5 | 12 | 23 | 28 | 10 | 120 |
30.5-35.5 | 14 | 37 | 33 | 5 | 70 |
35.5-40.5 | 26 | 63 | 38 | 0 | 0 |
40.5-45.5 | 12 | 75 | 43 | 5 | 60 |
45.5-50.5 | 16 | 91 | 48 | 10 | 160 |
50.5-55.5 | 9 | 100 | 53 | 15 | 135 |
=735 |
Now, N = 100, which is even.
The class interval containing or item is 35.5-40.5. Therefore, 35.5-40.5 is the median class.
We know,
Median
Here, l = 35.5, C = 37, f = 26, h = 5 and N = 100
Therefore, Median
Now, we calculate the absolute values of the deviations from median, and
= 735
Hence, the mean deviation about the median is 7.35
Class 11 maths chapter 15 ncert solutions - Exercise: 15.2
Question:1. Find the mean and variance for each of the data.
Answer:
Mean ( ) of the given data:
The respective values of the deviations from mean, are
-3, -2, 1 3 4 -5 -1 3
Hence, Mean = 9 and Variance = 9.25
Question:2. Find the mean and variance for each of the data.
First n natural numbers.
Answer:
Mean ( ) of first n natural numbers:
We know, Variance
We know that
Hence, Mean = and Variance =
Question:3. Find the mean and variance for each of the data
First 10 multiples of 3
Answer:
First 10 multiples of 3 are:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Mean ( ) of the above values:
The respective values of the deviations from mean, are
-13.5, -10.5, -7.5, -4.5, -1.5, 1.5, 4.5, 7.5, 10.5, 13.5
Hence, Mean = 16.5 and Variance = 74.25
Question:4. Find the mean and variance for each of the data.
Answer:
6 | 2 | 12 | -13 | 169 | 338 |
10 | 4 | 40 | -9 | 81 | 324 |
14 | 7 | 98 | -5 | 25 | 175 |
18 | 12 | 216 | -1 | 1 | 12 |
24 | 8 | 192 | 5 | 25 | 200 |
28 | 4 | 112 | 9 | 81 | 324 |
30 | 3 | 90 | 13 | 169 | 363 |
= 40 | = 760 | =1736 |
We know, Variance,
Hence, Mean = 19 and Variance = 43.4
Question:5. Find the mean and variance for each of the data.
Answer:
92 | 3 | 276 | -8 | 64 | 192 |
93 | 2 | 186 | -7 | 49 | 98 |
97 | 3 | 291 | -3 | 9 | 27 |
98 | 2 | 196 | -2 | 4 | 8 |
102 | 6 | 612 | 2 | 4 | 24 |
104 | 3 | 312 | 4 | 16 | 48 |
109 | 3 | 327 | 9 | 81 | 243 |
= 22 | = 2200 | =640 |
We know, Variance,
Hence, Mean = 100 and Variance = 29.09
Question:6 Find the mean and standard deviation using short-cut method.
Answer:
Let the assumed mean, A = 64 and h = 1
60 | 2 | -4 | 16 | -8 | 32 |
61 | 1 | -3 | 9 | -3 | 9 |
62 | 12 | -2 | 4 | -24 | 48 |
63 | 29 | -1 | 1 | -29 | 29 |
64 | 25 | 0 | 0 | 0 | 0 |
65 | 12 | 1 | 1 | 12 | 12 |
66 | 10 | 2 | 4 | 20 | 40 |
67 | 4 | 3 | 9 | 12 | 36 |
68 | 5 | 4 | 16 | 20 | 80 |
=100 | = 0 | =286 |
Mean,
We know, Variance,
We know, Standard Deviation =
Hence, Mean = 64 and Standard Deviation = 1.691
Question:âââââââ7. Find the mean and variance for the following frequency distributions.
Classes | 0-30 | 30-60 | 60-90 | 90-120 | 120-150 | 150-180 | 180-210 |
Frequencies | 2 | 3 | 5 | 10 | 3 | 5 | 2 |
Answer:
Classes | Frequency | Mid point | ||||
0-30 | 2 | 15 | 30 | -92 | 8464 | 16928 |
30-60 | 3 | 45 | 135 | -62 | 3844 | 11532 |
60-90 | 5 | 75 | 375 | -32 | 1024 | 5120 |
90-120 | 10 | 105 | 1050 | 2 | 4 | 40 |
120-150 | 3 | 135 | 405 | 28 | 784 | 2352 |
150-180 | 5 | 165 | 825 | 58 | 3364 | 16820 |
180-210 | 2 | 195 | 390 | 88 | 7744 | 15488 |
= N = 30 | = 3210 | =68280 |
We know, Variance,
Hence, Mean = 107 and Variance = 2276
Question:âââââââ8. Find the mean and variance for the following frequency distributions.
Classes | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
Frequencies | 5 | 8 | 15 | 16 | 6 |
Answer:
Classes | Frequency | Mid-point | ||||
0-10 | 5 | 5 | 25 | -22 | 484 | 2420 |
10-20 | 8 | 15 | 120 | -12 | 144 | 1152 |
20-30 | 15 | 25 | 375 | -2 | 4 | 60 |
30-40 | 16 | 35 | 560 | 8 | 64 | 1024 |
40-50 | 6 | 45 | 270 | 18 | 324 | 1944 |
= N = 50 | = 1350 | =6600 |
We know, Variance,
Hence, Mean = 27 and Variance = 132
Question:9. Find the mean, variance and standard deviation using short-cut method.
Height in cms | 70-75 | 75-80 | 80-85 | 85-90 | 90-95 | 95-100 | 100-105 | 105-110 | 110-115 |
No. of students | 3 | 4 | 7 | 7 | 15 | 9 | 6 | 6 | 3 |
Answer:
Let the assumed mean, A = 92.5 and h = 5
Height in cms | Frequency | Midpoint | ||||
70-75 | 3 | 72.5 | -4 | 16 | -12 | 48 |
75-80 | 4 | 77.5 | -3 | 9 | -12 | 36 |
80-85 | 7 | 82.5 | -2 | 4 | -14 | 28 |
85-90 | 7 | 87.5 | -1 | 1 | -7 | 7 |
90-95 | 15 | 92.5 | 0 | 0 | 0 | 0 |
95-100 | 9 | 97.5 | 1 | 1 | 9 | 9 |
100-105 | 6 | 102.5 | 2 | 4 | 12 | 24 |
105-110 | 6 | 107.5 | 3 | 9 | 18 | 54 |
110-115 | 3 | 112.5 | 4 | 16 | 12 | 48 |
=N = 60 | = 6 | =254 |
Mean,
We know, Variance,
We know, Standard Deviation =
Hence, Mean = 93, Variance = 105.583 and Standard Deviation = 10.275
Question:10. The diameters of circles (in mm) drawn in a design are given below:
Diameters | 33-36 | 37-40 | 41-44 | 45-48 | 49-52 |
No. of circles | 15 | 17 | 21 | 22 | 25 |
Calculate the standard deviation and mean diameter of the circles.
[ Hint First make the data continuous by making the classes as and then proceed.]
Answer:
Let the assumed mean, A = 92.5 and h = 5
Diameters | No. of circles | Midpoint | ||||
32.5-36.5 | 15 | 34.5 | -2 | 4 | -30 | 60 |
36.5-40.5 | 17 | 38.5 | -1 | 1 | -17 | 17 |
40.5-44.5 | 21 | 42.5 | 0 | 0 | 0 | 0 |
44.5-48.5 | 22 | 46.5 | 1 | 1 | 22 | 22 |
48.5-52.5 | 25 | 50.5 | 2 | 4 | 50 | 100 |
=N = 100 | = 25 | =199 |
Mean,
We know, Variance,
We know, Standard Deviation =
Hence, Mean = 43.5, Variance = 30.84 and Standard Deviation = 5.553
Statistics class 11 NCERT solutions - Exercise: 15.3
Question:1. From the data given below state which group is more variable, A or B?
Marks | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
Group A | 9 | 17 | 32 | 33 | 40 | 10 | 9 |
Group B | 10 | 20 | 30 | 25 | 43 | 15 | 7 |
Answer:
The group having a higher coefficient of variation will be more variable.
Let the assumed mean, A = 45 and h = 10
For Group A
Marks | Group A | Midpoint | ||||
10-20 | 9 | 15 | -3 | 9 | -27 | 81 |
20-30 | 17 | 25 | -2 | 4 | -34 | 68 |
30-40 | 32 | 35 | -1 | 1 | -32 | 32 |
40-50 | 33 | 45 | 0 | 0 | 0 | 0 |
50-60 | 40 | 55 | 1 | 1 | 40 | 40 |
60-70 | 10 | 65 | 2 | 4 | 20 | 40 |
70-80 | 9 | 75 | 3 | 9 | 27 | 81 |
=N = 150 | = -6 | =342 |
Mean,
We know, Variance,
We know, Standard Deviation =
Coefficient of variation =
C.V.(A) =
Similarly,
For Group B
Marks | Group A | Midpoint | ||||
10-20 | 10 | 15 | -3 | 9 | -30 | 90 |
20-30 | 20 | 25 | -2 | 4 | -40 | 80 |
30-40 | 30 | 35 | -1 | 1 | -30 | 30 |
40-50 | 25 | 45 | 0 | 0 | 0 | 0 |
50-60 | 43 | 55 | 1 | 1 | 43 | 43 |
60-70 | 15 | 65 | 2 | 4 | 30 | 60 |
70-80 | 7 | 75 | 3 | 9 | 21 | 72 |
=N = 150 | = -6 | =375 |
Mean,
We know, Variance,
We know, Standard Deviation =
Coefficient of variation =
C.V.(B) =
Since C.V.(B) > C.V.(A)
Therefore, Group B is more variable.
Question:2 From the prices of shares X and Y below, find out which is more stable in value:
X | 35 | 54 | 52 | 53 | 56 | 58 | 52 | 50 | 51 | 49 |
Y | 108 | 107 | 105 | 105 | 106 | 107 | 104 | 103 | 104 | 101 |
Answer:
X( ) | Y( ) | ||
35 | 108 | 1225 | 11664 |
54 | 107 | 2916 | 11449 |
52 | 105 | 2704 | 11025 |
53 | 105 | 2809 | 11025 |
56 | 106 | 8136 | 11236 |
58 | 107 | 3364 | 11449 |
52 | 104 | 2704 | 10816 |
50 | 103 | 2500 | 10609 |
51 | 104 | 2601 | 10816 |
49 | 101 | 2401 | 10201 |
=510 | = 1050 | =26360 | =110290 |
For X,
Mean ,
Variance,
We know, Standard Deviation =
C.V.(X) =
Similarly, For Y,
Mean ,
Variance,
We know, Standard Deviation =
C.V.(Y) =
Since C.V.(Y) < C.V.(X)
Hence Y is more stable.
Which firm A or B pays larger amount as monthly wages?
Answer:
Given, Mean of monthly wages of firm A = 5253
Number of wage earners = 586
Total amount paid = 586 x 5253 = 30,78,258
Again, Mean of monthly wages of firm B = 5253
Number of wage earners = 648
Total amount paid = 648 x 5253 = 34,03,944
Hence firm B pays larger amount as monthly wages.
Question:3(ii) An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results:
Which firm, A or B, shows greater variability in individual wages?
Answer:
Given, Variance of firm A = 100
Standard Deviation =
Again, Variance of firm B = 121
Standard Deviation =
Since , firm B has greater variability in individual wages.
Question:4 The following is the record of goals scored by team A in a football session:
No. of goals scored | 0 | 1 | 2 | 3 | 4 |
No. of matches | 1 | 9 | 7 | 5 | 3 |
For the team B, mean number of goals scored per match was 2 with a standard deviation goals. Find which team may be considered more consistent?
Answer:
No. of goals scored | Frequency | |||
0 | 1 | 0 | 0 | 0 |
1 | 9 | 1 | 9 | 9 |
2 | 7 | 4 | 14 | 28 |
3 | 5 | 9 | 15 | 45 |
4 | 3 | 16 | 12 | 48 |
=N = 25 | = 50 | =130 |
For Team A,
Mean,
We know, Variance,
We know, Standard Deviation =
C.V.(A) =
For Team B,
Mean = 2
Standard deviation, = 1.25
C.V.(B) =
Since C.V. of firm B is more than C.V. of A.
Therefore, Team A is more consistent.
Which is more varying, the length or weight?
Answer:
For lenght x,
Mean,
We know, Variance,
We know, Standard Deviation =
C.V.(x) =
For weight y,
Mean,
Mean,
We know, Variance,
We know, Standard Deviation =
C.V.(y) =
Since C.V.(y) > C.V.(x)
Therefore, weight is more varying.
Statistics maths chapter 15 class 11 - Miscellaneous Exercise
Answer:
Given,
The mean and variance of 8 observations are 9 and 9.25, respectively
Let the remaining two observations be x and y,
Observations: 6, 7, 10, 12, 12, 13, x, y.
∴ Mean,
60 + x + y = 72
x + y = 12 -(i)
Now, Variance
(Using (i))
-(ii)
Squaring (i), we get
(iii)
(iii) - (ii) :
2xy = 64 (iv)
Now, (ii) - (iv):
-(v)
Hence, From (i) and (v):
x – y = 4 x = 8 and y = 4
x – y = -4 x = 4 and y = 8
Therefore, The remaining observations are 4 and 8. (in no order)
Answer:
Given,
The mean and variance of 7 observations are 8 and 16, respectively
Let the remaining two observations be x and y,
Observations: 2, 4, 10, 12, 14, x, y
∴ Mean,
42 + x + y = 56
x + y = 14 -(i)
Now, Variance
(Using (i))
-(ii)
Squaring (i), we get
(iii)
(iii) - (ii) :
2xy = 96 (iv)
Now, (ii) - (iv):
-(v)
Hence, From (i) and (v):
x – y = 2 x = 8 and y = 6
x – y = -2 x = 6 and y = 8
Therefore, The remaining observations are 6 and 8. (in no order)
Answer:
Given,
Mean = 8 and Standard deviation = 4
Let the observations be
Mean,
Now, Let _{ } be the the resulting observations if each observation is multiplied by 3:
New mean,
= 24
We know that,
Standard Deviation =
-(i)
Now, Substituting the values of in (i):
Hence, the variance of the new observations =
Therefore, Standard Deviation = = = 12
Answer:
Given, Mean = and variance =
Now, Let _{ } be the the resulting observations if each observation is multiplied by a:
Hence the mean of the new observations is
We know,
Now, Substituting the values of :
Hence the variance of the new observations is
Hence proved.
If wrong item is omitted.
Answer:
Given,
Number of observations, n = 20
Also, Incorrect mean = 10
And, Incorrect standard deviation = 2
Thus, incorrect sum = 200
Hence, correct sum of observations = 200 – 8 = 192
Therefore, Correct Mean = (Correct Sum)/19
= 10.1
Now, Standard Deviation,
,which is the incorrect sum.
Thus, New sum = Old sum - (8x8)
= 2080 – 64
= 2016
Hence, Correct Standard Deviation =
=2.02
If it is replaced by
Answer:
Given,
Number of observations, n = 20
Also, Incorrect mean = 10
And, Incorrect standard deviation = 2
Thus, incorrect sum = 200
Hence, correct sum of observations = 200 – 8 + 12 = 204
Therefore, Correct Mean = (Correct Sum)/20
= 10.2
Now, Standard Deviation,
,which is the incorrect sum.
Thus, New sum = Old sum - (8x8) + (12x12)
= 2080 – 64 + 144
= 2160
Hence, Correct Standard Deviation =
= 1.98
Which of the three subjects shows highest variability in marks and which shows the lowest?
Answer:
Given,
Standard deviation of physics = 15
Standard deviation of chemistry = 20
Standard deviation of mathematics = 12
We know ,
C.V.
The subject with greater C.V will be more variable than others.
Hence, Mathematics has lowest variability and Chemistry has highest variability.
Answer:
Given,
Initial Number of observations, n = 100
Thus, incorrect sum = 2000
Hence, New sum of observations = 2000 - 21-21-18 = 1940
New number of observation, n' = 100-3 =97
Therefore, New Mean = New Sum)/100
= 20
Now, Standard Deviation,
,which is the incorrect sum.
Thus, New sum = Old (Incorrect) sum - (21x21) - (21x21) - (18x18)
= 40900 - 441 - 441 - 324
= 39694
Hence, Correct Standard Deviation =
= 3.036
X represents values or scores,
N represents the number of values or scores.
interested students can practice class 11 maths ch 15 question answer using the following exercises.
chapter-1 | |
chapter-2 | |
chapter-3 | |
chapter-4 | |
chapter-5 | |
chapter-6 | |
chapter-7 | |
chapter-8 | |
chapter-9 | |
chapter-10 | |
chapter-11 | |
chapter-12 | |
chapter-13 | |
chapter-14 | |
chapter-15 | Statistics |
chapter-16 |
Easy to Understand: The ch 15 maths class 11 solutions are written in simple language and are easy to understand, which makes them accessible for students who may not have a strong background in statistics.
Comprehensive Coverage: The statistics NCERT solutions cover all the important concepts and topics in Statistics Class 11 NCERT curriculum. This ensures that students are well-prepared and have a thorough understanding of the subject.
Structured Format: The ch 15 maths class 11 solutions are presented in a structured format that helps students to organize their thoughts and approach problems in a systematic manner.
This chapter seems very easy but it very useful in the field of science, research reliable predictions, or forecasts for future use, decision making, etc. There are 7 questions are given in the miscellaneous exercise. Try to solve them also to get command in this chapter. In NCERT solutions for class 11 maths chapter 15 statistics, you will get solutions to miscellaneous exercise too.
The concept of standard deviation involves measuring the amount of variation or deviation present in a given set of values, and the range is determined by considering the level of standard deviation. For a clearer understanding of this term, students are encouraged to download the PDF solutions provided by Careers360, which are available in both chapter-wise and exercise-wise formats to cater to their specific needs.
The topics discussed in class 11 maths statistics are
1. Introduction
2. Methods of Dispersion
3. Range
4. Mean Deviation
5. Variance and Standard Deviation
6. Analysis of Frequency Distributions
The NCERT Solutions for class 11 chapter 15 maths offer precise and straightforward explanations that aid students in achieving good grades in their exams. The solutions' methodical approach to problem-solving provides students with a clear understanding of the marks allocation as per the latest CBSE Syllabus 2023. By using these class 11th statistics solutions , students can identify their weak areas and work towards improving them, resulting in better academic performance.
Students can find a detailed NCERT solutions for class 11 maths by clicking on the link. they can practice these solutions to get confidence in the concepts and in-depth understanding that ultimately lead to high score in the exam.
Application Date:20 November,2023 - 19 December,2023
Application Date:20 November,2023 - 19 December,2023
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
A career as Bank Probationary Officer (PO) is seen as a promising career opportunity and a white-collar career. Each year aspirants take the Bank PO exam. This career provides plenty of career development and opportunities for a successful banking future. If you have more questions about a career as Bank Probationary Officer (PO), what is probationary officer or how to become a Bank Probationary Officer (PO) then you can read the article and clear all your doubts.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
A career as a Finance Executive requires one to be responsible for monitoring an organisation's income, investments and expenses to create and evaluate financial reports. His or her role involves performing audits, invoices, and budget preparations. He or she manages accounting activities, bank reconciliations, and payable and receivable accounts.
An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.
Bank Branch Managers work in a specific section of banking related to the invention and generation of capital for other organisations, governments, and other entities. Bank Branch Managers work for the organisations and underwrite new debts and equity securities for all type of companies, aid in the sale of securities, as well as help to facilitate mergers and acquisitions, reorganisations, and broker trades for both institutions and private investors.
Treasury analyst career path is often regarded as certified treasury specialist in some business situations, is a finance expert who specifically manages a company or organisation's long-term and short-term financial targets. Treasurer synonym could be a financial officer, which is one of the reputed positions in the corporate world. In a large company, the corporate treasury jobs hold power over the financial decision-making of the total investment and development strategy of the organisation.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.
A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.
A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
Individuals in the architecture career are the building designers who plan the whole construction keeping the safety and requirements of the people. Individuals in architect career in India provides professional services for new constructions, alterations, renovations and several other activities. Individuals in architectural careers in India visit site locations to visualize their projects and prepare scaled drawings to submit to a client or employer as a design. Individuals in architecture careers also estimate build costs, materials needed, and the projected time frame to complete a build.
Having a landscape architecture career, you are involved in site analysis, site inventory, land planning, planting design, grading, stormwater management, suitable design, and construction specification. Frederick Law Olmsted, the designer of Central Park in New York introduced the title “landscape architect”. The Australian Institute of Landscape Architects (AILA) proclaims that "Landscape Architects research, plan, design and advise on the stewardship, conservation and sustainability of development of the environment and spaces, both within and beyond the built environment". Therefore, individuals who opt for a career as a landscape architect are those who are educated and experienced in landscape architecture. Students need to pursue various landscape architecture degrees, such as M.Des, M.Plan to become landscape architects. If you have more questions regarding a career as a landscape architect or how to become a landscape architect then you can read the article to get your doubts cleared.
An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A veterinary doctor is a medical professional with a degree in veterinary science. The veterinary science qualification is the minimum requirement to become a veterinary doctor. There are numerous veterinary science courses offered by various institutes. He or she is employed at zoos to ensure they are provided with good health facilities and medical care to improve their life expectancy.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
When it comes to an operation theatre, there are several tasks that are to be carried out before as well as after the operation or surgery has taken place. Such tasks are not possible without surgical tech and surgical tech tools. A single surgeon cannot do it all alone. It’s like for a footballer he needs his team’s support to score a goal the same goes for a surgeon. It is here, when a surgical technologist comes into the picture. It is the job of a surgical technologist to prepare the operation theatre with all the required equipment before the surgery. Not only that, once an operation is done it is the job of the surgical technologist to clean all the equipment. One has to fulfil the minimum requirements of surgical tech qualifications.
Also Read: Career as Nurse
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
Are you searching for a chemical pathologist job description? A chemical pathologist is a skilled professional in healthcare who utilises biochemical laboratory tests to diagnose disease by analysing the levels of various components or constituents in the patient’s body fluid.
A Biochemical Engineer is a professional involved in the study of proteins, viruses, cells and other biological substances. He or she utilises his or her scientific knowledge to develop products, medicines or ways to improve quality and refine processes. A Biochemical Engineer studies chemical functions occurring in a living organism’s body. He or she utilises the observed knowledge to alter the composition of products and develop new processes. A Biochemical Engineer may develop biofuels or environmentally friendly methods to dispose of waste generated by industries.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages. Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.
If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.
They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.
Fashion bloggers use multiple social media platforms to recommend or share ideas related to fashion. A fashion blogger is a person who writes about fashion, publishes pictures of outfits, jewellery, accessories. Fashion blogger works as a model, journalist, and a stylist in the fashion industry. In current fashion times, these bloggers have crossed into becoming a star in fashion magazines, commercials, or campaigns.
Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.
The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.
Careers in videography are art that can be defined as a creative and interpretive process that culminates in the authorship of an original work of art rather than a simple recording of a simple event. It would be wrong to portrait it as a subcategory of photography, rather photography is one of the crafts used in videographer jobs in addition to technical skills like organization, management, interpretation, and image-manipulation techniques. Students pursue Visual Media, Film, Television, Digital Video Production to opt for a videographer career path. The visual impacts of a film are driven by the creative decisions taken in videography jobs. Individuals who opt for a career as a videographer are involved in the entire lifecycle of a film and production.
An SEO Analyst is a web professional who is proficient in the implementation of SEO strategies to target more keywords to improve the reach of the content on search engines. He or she provides support to acquire the goals and success of the client’s campaigns.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.
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Quality Assurance Manager Job Description: A QA Manager is an administrative professional responsible for overseeing the activity of the QA department and staff. It involves developing, implementing and maintaining a system that is qualified and reliable for testing to meet specifications of products of organisations as well as development processes.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment.
A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.
.NET Developer Job Description: A .NET Developer is a professional responsible for producing code using .NET languages. He or she is a software developer who uses the .NET technologies platform to create various applications. Dot NET Developer job comes with the responsibility of creating, designing and developing applications using .NET languages such as VB and C#.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
A DevOps Architect is responsible for defining a systematic solution that fits the best across technical, operational and and management standards. He or she generates an organised solution by examining a large system environment and selects appropriate application frameworks in order to deal with the system’s difficulties.
Individuals who are interested in working as a Cloud Administration should have the necessary technical skills to handle various tasks related to computing. These include the design and implementation of cloud computing services, as well as the maintenance of their own. Aside from being able to program multiple programming languages, such as Ruby, Python, and Java, individuals also need a degree in computer science.
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