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Mathematics · Class 11 Science

Ch 13Statistics — Class 11 Mathematics, concept-first.

Statistics deals with data collected for a specific purpose. We make decisions about the data by analysing and interpreting it. In earlier classes, you learned how to represent data graphically (bar graphs, histograms, frequency polygons) and in tabular form.

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Mean Deviation About Mean

Imagine you have a small set of numbers: the marks of five students in a test: 4, 6, 8, 10, 12. The average (mean) is 8. Now, each student is some distance away from this average.

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CA Foundation

1 concept · 4 appearances

2025, 2022

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

13.1

Introduction

Statistics deals with data collected for a specific purpose. We make decisions about the data by analysing and interpreting it.

13.2

Measures of Dispersion

When you have a set of data, the central tendency (mean, median, mode) tells you where the data clusters. But two different datasets can have the same mean yet look completely different.

13.3

Range

When we look at two sets of data, the first thing we often notice is how spread out the numbers are. In the example of batsmen A and B, we saw that A’s scores ranged from 0 to 117, while B’s scores we…

13.4

Mean Deviation

When we talk about the spread of a set of observations, a natural first thought is to look at how far each observation lies from some central value . The deviation of an observation from is simply .

13.4.1

Mean Deviation for Ungrouped Data

When we have observations , the mean deviation measures the average distance of each observation from a chosen central value. The central value we use is either the mean () or the median ().

+Worked Examplesi3 questions
  1. Example 1Find the mean deviation about the mean for the following data: 6, 7, 10, 12, 13, 4, 8, 12Free
  2. Example 2Find the mean deviation about the mean for the following data: 12, 3, 18, 17, 4, 9, 17, 19, 20, 15, 8, 17, 2, 3, 16, 11, 3, 1, 0, 5Preview
  3. Example 3Find the mean deviation about the median for the following data: 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21.Preview
13.4.2

Mean Deviation for Grouped Data

16 Q

Data can be grouped in two ways: as a discrete frequency distribution (where each observation is a distinct value with its own frequency) or as a continuous frequency distribution (where data is divid…

+Worked Examplesi4 questions
  1. Example 4Find the mean deviation about the mean for the following data: | $x_i$ | $f_i$ | |---|---| | 2 | 2 | | 5 | 8 | | 6 | 10 | | 8 | 7 | | 10 | 8…Free
  2. Example 5Find the mean deviation about the median for the following data: | $x_i$ | $f_i$ | |---|---| | 3 | 3 | | 6 | 4 | | 9 | 5 | | 12 | 2 | | 13 |…Free
  3. Example 6Find the mean deviation about the mean for the following data. | Marks obtained | Number of students | |---|---| | 10-20 | 2 | | 20-30 | 3 |…Preview
  4. Example 7Calculate the mean deviation about median for the following data: | Class | Frequency | |---|---| | 0-10 | 6 | | 10-20 | 7 | | 20-30 | 15 |…Preview
+Exercise 13.1i12 questions
  1. Q1Find the mean deviation about the mean for the following data: 4, 7, 8, 9, 10, 12, 13, 17Free
  2. Q2Find the mean deviation about the mean for the following data: 38, 70, 48, 40, 42, 55, 63, 46, 54, 44Free
  3. Q3Find the mean deviation about the median for the following data: 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17Free
  4. Q4Find the mean deviation about the median for the following data: 36, 72, 46, 42, 60, 45, 53, 46, 51, 49Preview
  5. Q5Find the mean deviation about the mean for the following data: | $x_i$ | $f_i$ | |---|---| | 5 | 7 | | 10 | 4 | | 15 | 6 | | 20 | 3 | | 25 |…Preview
  6. Q6Find the mean deviation about the mean for the following data: | $x_i$ | $f_i$ | |---|---| | 10 | 4 | | 30 | 24 | | 50 | 28 | | 70 | 16 | |…Preview
  7. Q7Find the mean deviation about the median for the following data: | $x_i$ | $f_i$ | |---|---| | 5 | 8 | | 7 | 6 | | 9 | 2 | | 10 | 2 | | 12 |…Preview
  8. Q8Find the mean deviation about the median for the following data: | $x_i$ | $f_i$ | |---|---| | 15 | 3 | | 21 | 5 | | 27 | 6 | | 30 | 7 | | 3…Preview
  9. Q9Find the mean deviation about the mean for the following data: | Income per day (in ₹) | Number of persons | |---|---| | 0-100 | 4 | | 100-2…Preview
  10. Q10Find the mean deviation about the mean for the following data: | Height in cms | Number of boys | |---|---| | 95-105 | 9 | | 105-115 | 13 |…Preview
  11. Q11Find the mean deviation about median for the following data: | Marks | Number of Girls | |---|---| | 0-10 | 6 | | 10-20 | 8 | | 20-30 | 14 |…Preview
  12. Q12Calculate the mean deviation about median age for the age distribution of 100 persons given below: | Age (in years) | Number | |---|---| | 1…Preview
13.4.3

Limitations of Mean Deviation

Mean deviation has two serious weaknesses that make it unreliable as a general-purpose measure of dispersion.

13.5

Variance and Standard Deviation

When we calculated the mean deviation, we took absolute values of the deviations to prevent positive and negative deviations from cancelling each other out.

13.5.1

Standard Deviation

When you calculate variance, you square the deviations . This squaring changes the units. If the original data is in centimetres, the variance is in square centimetres — a unit that has no direct phys…

+Worked Examplesi1 question
  1. Example 8Find the variance of the following data: 6, 8, 10, 12, 14, 16, 18, 20, 22, 24Preview
13.5.2

Standard Deviation of a Discrete Frequency Distribution

When data is presented as a discrete frequency distribution, each value occurs with a frequency . The total number of observations is , and the mean is .

+Worked Examplesi1 question
  1. Example 9Find the variance and standard deviation for the following data: | $x_i$ | $f_i$ | |---|---| | 4 | 3 | | 8 | 5 | | 11 | 9 | | 17 | 5 | | 20…Preview
13.5.3

Standard Deviation of a Continuous Frequency Distribution

When you have a continuous frequency distribution — data grouped into class intervals like 30–40, 40–50, and so on — you cannot directly use the raw values because you only know how many observations…

+Worked Examplesi2 questions
  1. Example 10Calculate the mean, variance and standard deviation for the following distribution: | Class | Frequency | |---|---| | 30-40 | 3 | | 40-50 |…Free
  2. Example 11Find the standard deviation for the following data: | $x_i$ | $f_i$ | |---|---| | 3 | 7 | | 8 | 10 | | 13 | 15 | | 18 | 10 | | 23 | 6 |Preview
13.5.4

Shortcut Method to Find Variance and Standard Deviation

11 Q

When the values of in a discrete distribution, or the mid-points of classes in a continuous distribution, are large numbers, the direct calculation of mean and variance becomes tedious and time-consum…

+Worked Examplesi1 question
  1. Example 12Calculate mean, variance and standard deviation for the following distribution. | Classes | Frequency | |---|---| | 30-40 | 3 | | 40-50 | 7…Preview
+Exercise 13.2i10 questions
  1. Q1Find the mean and variance for the following data: 6, 7, 10, 12, 13, 4, 8, 12Free
  2. Q2Find the mean and variance for the first $n$ natural numbers.Free
  3. Q3Find the mean and variance for the first 10 multiples of 3.Free
  4. Q4Find the mean and variance for the following data: | $x_i$ | $f_i$ | |---|---| | 6 | 2 | | 10 | 4 | | 14 | 7 | | 18 | 12 | | 24 | 8 | | 28 |…Preview
  5. Q5Find the mean and variance for the following data: | $x_i$ | $f_i$ | |---|---| | 92 | 3 | | 93 | 2 | | 97 | 3 | | 98 | 2 | | 102 | 6 | | 104…Preview
  6. Q6Find the mean and standard deviation using short-cut method. | $x_i$ | $f_i$ | |---|---| | 60 | 2 | | 61 | 1 | | 62 | 12 | | 63 | 29 | | 64…Preview
  7. Q7Find the mean and variance for the following frequency distribution. | Classes | Frequencies | |---|---| | 0-30 | 2 | | 30-60 | 3 | | 60-90…Preview
  8. Q8Find the mean and variance for the following frequency distribution. | Classes | Frequencies | |---|---| | 0-10 | 5 | | 10-20 | 8 | | 20-30…Preview
  9. Q9Find the mean, variance and standard deviation using short-cut method. | Height in cms | No. of children | |---|---| | 70-75 | 3 | | 75-80 |…Preview
  10. Q10The diameters of circles (in mm) drawn in a design are given below: | Diameters | No. of circles | |---|---| | 33-36 | 15 | | 37-40 | 17 | |…Preview

Miscellaneous Examples

+Miscellaneous Examplesi4 questions
  1. Example 13The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.Free
  2. Example 14The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.Free
  3. Example 15If each of the observation $x_1, x_2, \ldots, x_n$ is increased by '$a$', where $a$ is a negative or positive number, show that the variance…Preview
  4. Example 16The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead…Preview

Miscellaneous Exercise on Chapter 13

+Miscellaneous Exercisei6 questions
  1. Q1The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the r…Free
  2. Q2The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, 14. Find the remaining two…Free
  3. Q3The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and…Preview
  4. Q4Given that $\bar{x}$ is the mean and $\sigma^2$ is the variance of $n$ observations $x_1, x_2, \ldots, x_n$. Prove that the mean and varianc…Preview
  5. Q5The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8…Preview
  6. Q6The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three obs…Preview

Summary

- Measures of dispersion quantify the spread of data: range, mean deviation, variance, and standard deviation. - Range = maximum value minimum value; simplest but least reliable.

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 46 questions46 questions
  1. Q1Find the mean deviation about the mean of the distribution: | Size | 20 | 21 | 22 | 23 | 24 | |---|---|---|---|---|---| | Frequency | 6 | 4…Free
  2. Q2Find the mean deviation about the median of the following distribution: | Marks obtained | 10 | 11 | 12 | 14 | 15 | |---|---|---|---|---|---…Free
  3. Q3Calculate the mean deviation about the mean of the set of first $n$ natural numbers when $n$ is an odd number.Free
  4. Q4Calculate the mean deviation about the mean of the set of first $n$ natural numbers when $n$ is an even number.Preview
  5. Q5Find the standard deviation of the first $n$ natural numbers.Preview
  6. Q6The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results: Number of obse…Preview
  7. Q7The mean and standard deviation of a set of $n_1$ observations are $\overline{x}_1$ and $s_1$, respectively while the mean and standard devi…Preview
  8. Q8Two sets each of 20 observations, have the same standard deviation 5. The first set has a mean 17 and the second a mean 22. Determine the st…Preview
  9. Q9The frequency distribution: | $x$ | A | 2A | 3A | 4A | 5A | 6A | |---|---|---|---|---|---|---| | $f$ | 2 | 1 | 1 | 1 | 1 | 1 | where A is a…Preview
  10. Q10For the frequency distribution: | $x$ | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---| | $f$ | 4 | 9 | 16 | 14 | 11 | 6 | Find the st…Preview
  11. Q11There are 60 students in a class. The following is the frequency distribution of the marks obtained by the students in a test: | Marks | 0 |…Preview
  12. Q12The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 6…Preview
  13. Q13Mean and standard deviation of 100 items are 50 and 4, respectively. Find the sum of all the item and the sum of the squares of the items.Preview
  14. Q14If for a distribution $\sum(x-5) = 3$, $\sum(x-5)^2 = 43$ and the total number of item is 18, find the mean and standard deviation.Preview
  15. Q15Find the mean and variance of the frequency distribution given below: | $x$ | $1 \le x < 3$ | $3 \le x < 5$ | $5 \le x < 7$ | $7 \le x < 10$…Preview
  16. Q16Calculate the mean deviation about the mean for the following frequency distribution: | Class interval | 0 - 4 | 4 - 8 | 8 - 12 | 12 - 16 |…Preview
  17. Q17Calculate the mean deviation from the median of the following data: | Class interval | 0 - 6 | 6 - 12 | 12 - 18 | 18 - 24 | 24 - 30 | |---|-…Preview
  18. Q18Determine the mean and standard deviation for the following distribution: | Marks | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |…Preview
  19. Q19The weights of coffee in 70 jars is shown in the following table: | Weight (in grams) | Frequency | |---|---| | 200 - 201 | 13 | | 201 - 202…Preview
  20. Q20Determine mean and standard deviation of first $n$ terms of an A.P. whose first term is $a$ and common difference is $d$.Preview
  21. Q21Following are the marks obtained, out of 100, by two students Ravi and Hashina in 10 tests. | Ravi | 25 | 50 | 45 | 30 | 70 | 42 | 36 | 48 |…Preview
  22. Q22Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations wer…Preview
  23. Q23While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the me…Preview
  24. Q24The mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from the mean is (A) 2 (B) 2.57 (C) 3 (D) 3.75Preview
  25. Q25Mean deviation for $n$ observations $x_1, x_2, \ldots, x_n$ from their mean $\overline{x}$ is given by (A) $\sum_{i=1}^{n}(x_i - \overline{x…Preview
  26. Q26When tested, the lives (in hours) of 5 bulbs were noted as follows: 1357, 1090, 1666, 1494, 1623 The mean deviations (in hours) from their m…Preview
  27. Q27Following are the marks obtained by 9 students in a mathematics test: 50, 69, 20, 33, 53, 39, 40, 65, 59 The mean deviation from the median…Preview
  28. Q28The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is (A) $\sqrt{\dfrac{52}{7}}$ (B) $\dfrac{52}{7}$ (C) $\sqrt{6}$ (D) 6Preview
  29. Q29Let $x_1, x_2, \ldots, x_n$ be $n$ observations and $\overline{x}$ be their arithmetic mean. The formula for the standard deviation is given…Preview
  30. Q30The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is (A) 50000 (B) 250000…Preview
  31. Q31Let $a, b, c, d, e$ be the observations with mean $m$ and standard deviation $s$. The standard deviation of the observations $a + k, b + k,…Preview
  32. Q32Let $x_1, x_2, x_3, x_4, x_5$ be the observations with mean $m$ and standard deviation $s$. The standard deviation of the observations $kx_1…Preview
  33. Q33Let $x_1, x_2, \ldots x_n$ be $n$ observations. Let $w_i = lx_i + k$ for $i = 1, 2, \ldots n$, where $l$ and $k$ are constants. If the mean…Preview
  34. Q34Standard deviations for first 10 natural numbers is (A) 5.5 (B) 3.87 (C) 2.97 (D) 2.87Preview
  35. Q35Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1 is added to each number, the variance of the numbers so obtained is (A) 6.5 (B) 2.8…Preview
  36. Q36Consider the first 10 positive integers. If we multiply each number by $-1$ and then add 1 to each number, the variance of the numbers so ob…Preview
  37. Q37The following information relates to a sample of size 60: $\sum x^2 = 18000$, $\sum x = 960$ The variance is (A) 6.63 (B) 16 (C) 22 (D) 44Preview
  38. Q38Coefficient of variation of two distributions are 50 and 60, and their arithmetic means are 30 and 25 respectively. Difference of their stan…Preview
  39. Q39The standard deviation of some temperature data in °C is 5. If the data were converted into ºF, the variance would be (A) 81 (B) 57 (C) 36 (…Preview
  40. Q40Coefficient of variation $= \dfrac{\ldots}{\text{Mean}} \times 100$Preview
  41. Q41If $\overline{x}$ is the mean of $n$ values of $x$, then $\sum_{i=1}^{n}(x_i - \overline{x})$ is always equal to _______. If $a$ has any val…Preview
  42. Q42If the variance of a data is 121, then the standard deviation of the data is _______.Preview
  43. Q43The standard deviation of a data is ___________ of any change in orgin, but is _____ on the change of scale.Preview
  44. Q44The sum of the squares of the deviations of the values of the variable is _______ when taken about their arithmetic mean.Preview
  45. Q45The mean deviation of the data is _______ when measured from the median.Preview
  46. Q46The standard deviation is _______ to the mean deviation taken from the arithmetic mean.Preview