Class 11 NCERT Solutions Chapter 15 Statistics Exercise 15.1 (original) (raw)

Last Updated : 18 Sep, 2024

The mean deviation is a measure of dispersion that gives us an idea of how far each data point in a set is from the mean or median of the dataset. This blog will guide you through calculating the mean deviation between the mean and the median for various datasets. Let's explore the step-by-step process for each dataset

Chapter 15 of Class 11 NCERT Mathematics deals with Statistics, a branch of mathematics that involves the collection, analysis, interpretation, and presentation of data. Exercise 15.1 specifically focuses on measures of central tendency, particularly the mean (arithmetic average) for ungrouped data.

Question 1. 4, 7, 8, 9, 10, 12, 13, 17

**Solution:

Given observations are = 4, 7, 8, 9, 10, 12, 13, 17

Therefore, Total Number of observations = n = 8

**Step 1: Calculating mean(a) for given data about which we have to find the mean deviation.

Mean(a) = ∑(all data points) / Total number of data points

= (4 + 7 + 8 + 9 + 10 + 12 + 13 + 17) / 8

= 80 / 8

a = 10

**Step 2: Finding deviation of each data point from mean(xi - a)

**x i	4	7	8	9	10	12	13	17
**x i - a	4 - 10 = -6	7 - 10 = -3	8 - 10 = -2	9 - 10 = -1	10 - 10 = 0	12 - 10 = 2	13 - 10 = 3	17 - 10 = 7

**Step 3: Taking absolute value of deviations, we get

|xi - a| = 6, 3, 2, 1, 0, 2, 3, 7

**Step 4: Required mean deviation about the mean is

M.D (a) = ∑ (|xi - a|) / n

= (6 + 3 + 2 + 1 + 0 + 2 + 3 + 7) / 8

= 24 / 8

= 3

_So, mean deviation for given observations is **3

Question 2. 38, 70, 48, 40, 42, 55, 63, 46, 54, 44

**Solution:

Given observations are = 38, 70, 48, 40, 42, 55, 63, 46, 54, 44

Therefore, Total Number of observations = n = 10

**Step 1: Calculating mean(a) for given data about which we have to find the mean deviation.

Mean(a) = ∑(all data points)/ Total number of data points

= (38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44)/ 10

= 500 / 10

a = 50

**Step2 : Finding deviation of each data point from mean(xi - a)

**x i	38	70	48	40	42	55	63	46	54	44
**x i - a	38 - 50= -12	70 - 50 = 20	48 - 50 = -2	40 - 50 = -10	42 - 50 = -8	55 - 50 = 5	63 - 50 = 13	46 - 50 = -4	54 - 50 = 4	44 - 50 = -6

**Step 3: Taking absolute value of deviations, we get

|xi - a| = 12, 20, 2, 10, 8, 5, 13, 4, 4, 6

**Step 4: Required mean deviation about the mean is

M.D(a)= ∑(|xi - a|) / n

= (12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6) / 10

= 84 / 10

= 8.4

_So, mean deviation for given observations is **8.4

Find the mean deviation about the median for the data in questions 3 and 4.

Question 3. 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

**Solution:

Given observations are = 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

Therefore, Total Number of observations = n = 12

**Step 1: Calculating median(M) for given data about which we have to find the mean deviation.

To calculate median for given data we have to arrange observations in ascending order

10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18

As number of observations are even median is given by following formula

Median = [(n/2)th observation+ ((n/2) + 1)th observation] / 2

(n/2)th observation = 12 / 2 = 6th observation= 13

((n/2)+1)th observation = (12/2) + 1= 7thobservation = 14

Therefore, Median = (13 + 10) / 2 = 13.5

**Step 2: Finding deviation of each data point from median(xi - M)

**x i	10	11	11	12	13	13	14	16	16	17	17	18
**x i -M	10 - 13.5 = -3.5	11 - 13.5 = -2.5	11 - 13.5 = -2.5	12 - 13.5 = -1.5	13 - 13.5 = -0.5	13 - 13.5 = 0.5	14 - 13.5 = 0.5	16 - 13.5 = 2.5	16 - 13.5 = 2.5	17 - 13.5 = 3.5	17 - 13.5 = 3.5	18 - 13.5 = 4.5

**Step 3: Taking absolute values of deviations we get

|xi - M| = 3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5

**Step 4: Required mean deviation about median is given by

M.D (M) = ∑(|xi - M|) / n

= (3.5 + 2.5 + 2.5 + 1.5 + 0.5 + 0.5 + 0.5 + 2.5 + 2.5 + 3.5 + 3.5 + 4.5) / 12

= 28 / 12

= 2.33

_So, mean deviation about median for given observations is **2.33

Question 4. 36, 72, 46, 42, 60, 45, 53, 46, 51, 49

**Solution:

Given observations are = 36, 72, 46, 42, 60, 45, 53, 46, 51, 49

Therefore, Total Number of observations = n = 10

**Step 1: Calculating median(M) for given data about which we have to find the mean deviation.

To calculate median for given data we have to arrange observations in ascending order

36, 42, 45, 46, 46, 49, 51, 53, 60, 72

As number of observations are even median is given by following formula

Median = [(n/2)th observation+ ((n/2) + 1)th observation] / 2

(n/2)th observation = 10 / 2 = 5th observation= 46

((n/2) + 1)th observation = (10/2) + 1 = 6th observation = 49

Therefore, Median = (46 + 49) / 2 = 47.5

**Step 2: Finding deviation of each data point from median(xi -M)

**x i	36	42	45	46	46	49	51	53	60	72
**x i - M	-11.5	-5.5	-2.5	-1.5	-1.5	1.5	3.5	5.5	12.5	24.5

**Step 3: Taking absolute values of deviations we get

|xi - M| = 11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5

**Step 4: Required mean deviation about median is given by

M.D (M) = ∑(|xi - M|) / n

= (11.5 + 5.5 + 2.5 + 1.5 + 1.5 + 1.5 + 3.5 + 5.5 + 12.5 + 24.5) / 10

= 70 / 10

= 7

_So, mean deviation about median for given observations is 7

Find the mean deviation about the mean for the data in questions 5 and 6.

Question 5.

**Solution:

Given table data is of discrete frequency distribution as

we have n = 5 distinct values(xi) along with their frequencies(fi).

**Step 1: Let us make a table of given data and append other columns after calculation

| **x i | **f i | **x i *** f i | **|x i **- a| | **f i *** |x i **- a| | | ------------- | ------------- | ------------------------------ | --------------------------- | -------------------------------------------- | | 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 | | **Total | 25 | 350 | | 158 |

**Step2 : Now to find the mean we have to first calculate the sum of given data

N = ∑ fi = (7 + 4 + 6 + 3 + 5) = 25

∑ xi * fi = (35 + 40 + 90 + 60 + 125) = 350

**Step 3: Find the mean using following formula

Mean (a) = ∑(xi * fi)/ N = 350 / 25 = 14

**Step 4: Using above mean find absolute values of deviations

from mean i.e |xi - a| and also find values of fi *|xi -a| column.

Now,

∑ fi *|xi - a| = (63 + 16 + 6 + 18 + 55) = 158

Using formula, M.D (a) = ∑(fi * |xi - a|)/ N

= 158 / 25

= 6.32

_So, mean deviation for given observations is 6.32

Question 6.

**Solution:

Given table data is of discrete frequency distribution as

we have n = 5 distinct values(xi) along with their frequencies(fi).

**Step 1: Let us make a table of given data and append other columns after calculation

| **x i | **f i | **x i *** f i | **|x i **- a| | **f i *** |x i **- a| | | ------------- | ------------- | ------------------------------ | --------------------------- | -------------------------------------------- | | 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 | | **Total | 80 | 4000 | | 1280 |

**Step 2: Now to find the mean we have to first calculate the sum of given data. From the above table,

N = ∑(fi) = 80 & ∑(xi * fi) = 4000

**Step 3: Find the mean using following formula

Mean (a) = ∑ (xi * fi)/ N = 4000 / 80 = 50

**Step 4: Using above mean find absolute values of deviations from mean i.e |xi - a| and also find values

of fi * |xi -a| column.

Now, from above table,

∑ fi *|xi - a| = (160 + 480 + 0 + 320 + 320) = 1280

Using formula, M.D (a) = ∑ fi *|xi -a| / N

= 1280 / 80

= 16

_So, mean deviation for given observations is **16

Find the mean deviation about the median for the data in question 7 and 8.

Question 7.

**Solution:

For the given discrete frequency distribution we have to find a mean deviation about the median.

**Step 1: Let us make a table of given data and append another column of cumulative frequencies.

The given observations are already in ascending order.

| **x i | **f i | **C.F | **|x i **- M| | **f i *** |x i **- M| | | ------------- | ------------- | --------- | --------------------------- | -------------------------------------------- | | 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 | | **Total | 26 | | | 84 |

**Step 2: Identifying the observation whose cumulative frequency is equal

to or just greater than N / 2 and then Finding the median.

N = ∑fi = 26 is even. We divide N by 2. Thus, 26/2 = 13

The cumulative frequency for greater than 13 is 14, for which corresponding observation is 7

Median = [(N/2)th observation + ((N/2) + 1)th observation] / 2

= (13th observation + 14th observation) / 2

= (7 + 7) / 2 = **7

**Step 3: Now, find absolute values of the deviations from median,

i.e., |xi - M| and fi * |xi - M| as shown in above table.

∑ fi * |xi - M| = (16 + 4 + 6 + 10 + 48) = 84

Using formula, M.D (M) = ∑ (fi * |xi - M|)/ N

= 84 / 26

= 3.23

So, mean deviation about median for given observations is **3.23

Question 8.

**Solution:

For the given discrete frequency distribution we have to find a mean deviation about the median.

**Step 1: Let us make a table of given data and append another column of cumulative frequencies.

The given observations are already in ascending order.

| **x i | **f i | **C.F | **|x i **- M| | **f i *** |x i **- M| | | ------------- | ------------- | --------- | --------------------------- | -------------------------------------------- | | 15 | 3 | 3 | 15 | 45 | | 21 | 5 | 8 | 9 | 45 | | 27 | 6 | 14 | 3 | 18 | | 30 | 7 | 21 | 0 | 0 | | 35 | 8 | 29 | 5 | 40 | | **Total | 29 | | | 148 |

**Step 2: Identifying the observation whose cumulative frequency is equal to or

just greater than N / 2 and then Finding the median.

Here, N = ∑fi = 29 is odd we divide N by 2. Thus, 29 / 2 = 14.5

The cumulative frequency for greater than 14.5 is 21, for which corresponding observation is 30

Therefore, Median = [(N/2)th observation+ ((N/2) + 1)th observation] / 2

= (15th observation + 16th observation) / 2

= (30 + 30)/2 = 30

**Step 3: Now, find absolute values of the deviations from median,

i.e., |xi - M| and fi * |xi - M| as shown in above table .

∑fi * |xi - M| = 148

Using formula, M.D (M) = ∑(fi * |xi - M|) / N

= 148 / 29

= 5.1

So, mean deviation about median for given observations is**5.1

Find the mean deviation about the mean for the following data in questions 9 and 10.

Question 9.

**Solution:

Given data is in continuous intervals along with their frequencies so it is in the

continuous frequency distribution. In this case, we assume that the frequency

in each class is centered at its mid-point.

**Step 1: So, we find a midpoint for each interval.

Then we append other columns similar to discrete frequency distribution.

| **Income per day in Rs. | **Number Of Person f i | **Midpoints x i | **f i x i | **|x i **- a| | **f i *** |x i **- a| | | ------------------------------- | ---------------------------------- | --------------------------- | ------------------------- | --------------------------- | -------------------------------------------- | | 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 | 1160 | | 700-800 | 3 | 750 | 2250 | 392 | 1176 | | **Total | 50 | | 17900 | | 7896 |

Step 2: Finding the sum of frequencies fi's and sum of fixi's

N = ∑fi = 50

∑fixi = 17900

Then mean of given data is given by

a = ∑ (fixi)/ N

= 17900 / 50

a = 358

**Step 3: Computing sum of column fi * |xi - a|

∑fi * |xi - a| = 7896

Thus mean deviation about mean is given by

M.D (a) = ∑fi * |xi - a| / N

= 7896 / 50

= 157.92

_Therefore, mean deviation about mean for given data is **157.92

Question 10.

**Solution:

Given data is in the continuous frequency distribution.

**Step 1: We find a midpoint for each interval and then append other columns.

| Height in cms | **Number of boys | **Midpoints x i | **f i x i | **|x i **- a| | **f i *** |x i **- a| | | --------------------- | ------------------------ | --------------------------- | ------------------------- | --------------------------- | -------------------------------------------- | | 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 | | **Total | 100 | | 12530 | | 1128.8 |

**Step 2: Finding the sum of frequencies fi's and the sum of fixi's

N = ∑fi = 100

∑ fixi = 17900

Then mean of given data is given by

a = ∑(fixi) / N

= 12530 / 100

a = 125.3

**Step 3: Computing sum of column fi * |xi - a|

∑fi * |xi - a| = 1128.8

Thus mean deviation about mean is given by

M.D(a) = ∑(fi * |xi - a|)/ N

= 1128.8 / 100

= 11.288

_Therefore, the mean deviation about mean for given data is **11.288

Find the mean deviation about the median for the following data in questions 11 and 12.

Question 11.

**Solution:

Given data is in the continuous frequency distribution &

here the only difference is that we have to calculate the median.

**Step 1: First we have to compute cumulative frequencies, then we find a midpoint for each interval and then append other columns.

The data is already arranged in ascending order

| **Marks | **Number of Girls f i | **C.F | **Midpoints x i | **|x i **- M| | **f i *** |x i **- M| | | ----------- | --------------------------------- | --------- | --------------------------- | --------------------------- | -------------------------------------------- | | 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 | | **Total | 50 | | | | 517.1 |

**Step 2: First identifying the interval in which median lies and then applying the formula to compute median

The class interval whose cumulative frequency is greater than equal to N/2 = 25 is 20 - 30.

So, 20 - 30 is the median class.

Then applying the formula

**Median (M) = l + {[(N / 2) - C] / f} * h

where, l = lower limit of the median class

h = width of median class

N = sum of frequencies

C = cumulative frequency of the class just preceding the median class

Therefore,

M = 20 + {[(25 - 14) / 14] * 10}

M = 27.85

**Step 3: Finding absolute values of the deviations from median as shown in table

computing sum of column fi * |xi - M|

∑fi * |xi - M| = 517.1

Thus, mean deviation about median is given by

M.D(M) = ∑fi * |xi - M| / N

= 517.1 / 50

= 10.34

_So, mean deviation about median for given observations is **10.34

Question12. Calculate the mean deviation about median age for the age distribution of 100 persons given below:

**Solution:

Converting the given data into continuous frequency distribution by subtracting 0.5

from the lower limit and adding 0.5 to the upper limit of each class interval.

Check if data is arranged in ascending order.

**Step 1: Finding midpoints and C.Fs and then appending other columns

| **Age (in years) | **Number f i | **C.F | **Midpoints x i | **|x i **- M| | **f i *** |x i - M| | | ------------------------ | ------------------------ | --------- | --------------------------- | --------------------------- | ------------------------------------------ | | 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 | 71 | 43 | 5 | 60 | | 45.5-50.5 | 16 | 95 | 48 | 10 | 160 | | 50.5-55.5 | 9 | 100 | 53 | 15 | 135 | | **Total | 100 | | | | 735 |

**Step 2: First identifying the interval in which median lies and then applying the formula to compute median

The class interval whose cumulative frequency is greater than equal to N/2 = 50 is 35.5-40.5.

So, 35.5-40.5 is the median class.

Then applying the formula

**Median (M) = l + {[(N / 2) - C] / f} * h

where l = lower limit of the median class = 35.5

h = width of median class = 5

N = sum of frequencies = 100

C = cumulative frequency of the class just preceding the median class = 37

f = frequency = 26

Therefore,

M = 35.5 + {[(50 - 37) / 26] * 5}

M = 38

**Step 3: Finding absolute values of the deviations from median as shown in table

computing sum of column fi * |xi - M|

∑fi * |xi - M| = 735

Thus, mean deviation about median is given by

M.D(M) = ∑fi * |xi - M| / N

= 735 / 100

= 7.35

_So, mean deviation about median for given observations is **7.35

Summary

Exercise 15.1 covers the calculation of arithmetic mean for ungrouped data. It introduces students to various methods of calculating the mean, including the direct method and the assumed mean method. The problems in this exercise help students understand how to compute the mean for different types of datasets and interpret the results in real-world contexts.