### Statistics Exercise 01

## Question:

Find the mean deviation about the mean for the data Height of cms Number of boys 95-105 9 105-115 13 115-125 26 125-135 30 135-145 12 145-155 10

## Answer:

Step 1: Find the mean of the data.

Mean = (95 + 105 + 115 + 125 + 135 + 145 + 155) ÷ 7 Mean = 115

Step 2: Find the deviation of each data point from the mean.

95-115 = -20 105-115 = -10 115-115 = 0 125-115 = 10 135-115 = 20 145-115 = 30 155-115 = 40

Step 3: Find the mean of the deviations.

Mean deviation = (-20 + -10 + 0 + 10 + 20 + 30 + 40) ÷ 7 Mean deviation = 10

## Question:

Find the mean deviation about the mean for the data: 4,7,8,9,10,12,13,17

## Answer:

Step 1: Find the mean of the data.

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

Mean = 10

Step 2: Find the deviation of each data point from the mean.

4: 4 - 10 = -6 7: 7 - 10 = -3 8: 8 - 10 = -2 9: 9 - 10 = -1 10: 10 - 10 = 0 12: 12 - 10 = 2 13: 13 - 10 = 3 17: 17 - 10 = 7

Step 3: Find the absolute value of each deviation.

|-6| = 6 |-3| = 3 |-2| = 2 |-1| = 1 |0| = 0 |2| = 2 |3| = 3 |7| = 7

Step 4: Find the mean of the absolute values.

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

Mean Deviation = 3.375

## Question:

Calculate the mean deviation about median age for the age distribution of 100 persons given below : Age Number 16-20 5 21-25 6 26-30 12 31-35 14 36-40 26 41-45 12 46-50 16 51-55 9

## Answer:

Step 1: Calculate the median age of the given age distribution.

Median Age = (31 + 36) / 2 = 33.5

Step 2: Calculate the absolute deviation of each age group from the median age.

Age Group Number Absolute Deviation 16-20 5 17.5 21-25 6 12.5 26-30 12 6.5 31-35 14 4.5 36-40 26 2.5 41-45 12 8.5 46-50 16 4.5 51-55 9 11.5

Step 3: Calculate the sum of the absolute deviations.

Sum of Absolute Deviations = 17.5 + 12.5 + 6.5 + 4.5 + 2.5 + 8.5 + 4.5 + 11.5 = 77

Step 4: Calculate the mean deviation about median age.

Mean Deviation about Median Age = Sum of Absolute Deviations / Total Number of Persons = 77 / 100 = 0.77

## Question:

Find the mean deviation about the median for the data 36,72,46,42,60,45,53,46,51,49

## Answer:

Step 1: Arrange the data in ascending order: 36, 42, 45, 46, 46, 49, 51, 53, 60, 72

Step 2: Find the median of the data: 46

Step 3: Calculate the deviation of each data point from the median: -10, -4, -1, 0, 0, 3, 5, 7, 14, 26

Step 4: Find the mean of the deviation: (-10 + -4 + -1 + 0 + 0 + 3 + 5 + 7 + 14 + 26) / 10 = 4.1

## Question:

Find the mean deviation about the mean for the data 38,70,48,40,42,55,63,46,54,44

## Answer:

Step 1: Find the mean of the data.

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

Mean = 48.2

Step 2: Subtract the mean from each data point.

38 - 48.2 = -10.2

70 - 48.2 = 21.8

48 - 48.2 = -0.2

40 - 48.2 = -8.2

42 - 48.2 = -6.2

55 - 48.2 = 6.8

63 - 48.2 = 14.8

46 - 48.2 = -2.2

54 - 48.2 = 5.8

44 - 48.2 = -4.2

Step 3: Take the absolute value of each difference.

|-10.2| = 10.2

|21.8| = 21.8

|-0.2| = 0.2

|-8.2| = 8.2

|-6.2| = 6.2

|6.8| = 6.8

|14.8| = 14.8

|-2.2| = 2.2

|5.8| = 5.8

|-4.2| = 4.2

Step 4: Add the absolute values together.

10.2 + 21.8 + 0.2 + 8.2 + 6.2 + 6.8 + 14.8 + 2.2 + 5.8 + 4.2 = 85

Step 5: Divide the sum of the absolute values by the number of data points.

85 / 10 = 8.5

Step 6: The mean deviation about the mean is 8.5.

## Question:

Find the mean deviation about the median for the data 13,17,16,14,11,13,10,16,11,18,12,17

## Answer:

Step 1: Arrange the data in order from least to greatest: 10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18

Step 2: Find the median of the data: 13

Step 3: Subtract the median from each data point: -3, -2, -2, -1, 0, 0, 1, 3, 3, 4, 4, 5

Step 4: Take the absolute value of each result: 3, 2, 2, 1, 0, 0, 1, 3, 3, 4, 4, 5

Step 5: Add up the absolute values: 21

Step 6: Divide the sum by the number of data points: 21/12 = 1.75

Therefore, the mean deviation about the median is 1.75.

## Question:

Find the mean deviation about the mean for the data in Exercises 5 and 6. xi 5 10 15 20 25 fi 7 4 6 3 5

## Answer:

Step 1: Find the mean (x̅): x̅ = (5 + 10 + 15 + 20 + 25)/5 x̅ = 15

Step 2: Find the deviation of each value (xi - x̅): 5 - 15 = -10 10 - 15 = -5 15 - 15 = 0 20 - 15 = 5 25 - 15 = 10

Step 3: Find the absolute value of the deviations: |-10| = 10 |-5| = 5 |0| = 0 |5| = 5 |10| = 10

Step 4: Multiply each absolute value by the frequency (fi): 10 x 7 = 70 5 x 4 = 20 0 x 6 = 0 5 x 3 = 15 10 x 5 = 50

Step 5: Add all of the products together: 70 + 20 + 0 + 15 + 50 = 155

Step 6: Divide the sum of the products by the total frequency (N): 155/20 = 7.75

The mean deviation about the mean is 7.75.

## Question:

Find the mean deviation about the mean for the data in xi 10 30 50 70 90 fi 4 24 28 16 8

## Answer:

Step 1: Calculate the mean (x̅) of the data:

x̅ = (10 + 30 + 50 + 70 + 90)/5

x̅ = 40

Step 2: Calculate the deviation of each data point from the mean:

xi - x̅

10 - 40 = -30 30 - 40 = -10 50 - 40 = 10 70 - 40 = 30 90 - 40 = 50

Step 3: Square each deviation:

(-30)2 = 900 (-10)2 = 100 102 = 100 302 = 900 502 = 2500

Step 4: Multiply each squared deviation by its frequency:

900 x 4 = 3600 100 x 24 = 2400 100 x 28 = 2800 900 x 16 = 14400 2500 x 8 = 20000

Step 5: Sum the products:

3600 + 2400 + 2800 + 14400 + 20000 = 43200

Step 6: Divide the sum by the total frequency (N):

43200/5 = 8640

Step 7: Take the square root of the result:

√8640 = 92.8

Therefore, the mean deviation about the mean for the given data is 92.8.

## Question:

Find the mean deviation from the mean for the following data: xi 5 7 9 10 12 15 fi 8 6 2 2 2 6

## Answer:

Step 1: Find the mean of the data.

Mean = (5*8 + 7*6 + 9*2 + 10*2 + 12*2 + 15*6) / (8 + 6 + 2 + 2 + 2 + 6)

Mean = 10

Step 2: Find the deviation of each data point from the mean.

Deviation of x1 = 5 - 10 = -5 Deviation of x2 = 7 - 10 = -3 Deviation of x3 = 9 - 10 = -1 Deviation of x4 = 10 - 10 = 0 Deviation of x5 = 12 - 10 = 2 Deviation of x6 = 15 - 10 = 5

Step 3: Find the mean of the deviations.

Mean Deviation = (-5 + -3 + -1 + 0 + 2 + 5) / 6

Mean Deviation = 0.5

## Question:

Find the mean deviation from the median for the following data: xi 15 21 27 30 fi 3 5 6 7

## Answer:

Step 1: Find the median of the data.

The median is 24.5, which is the average of 21 and 27.

Step 2: Find the absolute value of the difference between each value and the median.

15 - 24.5 = -9.5 21 - 24.5 = -3.5 27 - 24.5 = 2.5 30 - 24.5 = 5.5

Step 3: Multiply each absolute value by the corresponding frequency.

-9.5 x 3 = -28.5 -3.5 x 5 = -17.5 2.5 x 6 = 15 5.5 x 7 = 38.5

Step 4: Add all of the products together.

-28.5 + -17.5 + 15 + 38.5 = 8.5

Step 5: Divide the sum of the products by the total frequency.

8.5 / 21 = 0.4

The mean deviation from the median is 0.4.

## Question:

Find the mean deviation about the mean for the data: Income per day Numbers of persons 0-100 4 100-200 8 200-300 9 300-400 10 400-500 7 500-600 5 600-700 4 700-800 3

## Answer:

Step 1: Find the mean of the data.

Mean = (4*0 + 8*100 + 9*200 + 10*300 + 7*400 + 5*500 + 4*600 + 3*700)/(4+8+9+10+7+5+4+3)

Mean = (0 + 800 + 1800 + 3000 + 2800 + 2500 + 2400 + 2100)/(4+8+9+10+7+5+4+3)

Mean = 14300/(4+8+9+10+7+5+4+3)

Mean = 14300/47

Mean = 303.19

Step 2: Find the deviation of each value from the mean.

Deviation of 0-100 = (0 - 303.19) = -303.19

Deviation of 100-200 = (100 - 303.19) = -203.19

Deviation of 200-300 = (200 - 303.19) = -103.19

Deviation of 300-400 = (300 - 303.19) = -3.19

Deviation of 400-500 = (400 - 303.19) = 96.81

Deviation of 500-600 = (500 - 303.19) = 196.81

Deviation of 600-700 = (600 - 303.19) = 296.81

Deviation of 700-800 = (700 - 303.19) = 396.81

Step 3: Find the mean deviation about the mean.

Mean Deviation about the mean = (|-303.19| + |-203.19| + |-103.19| + |-3.19| + |96.81| + |196.81| + |296.81| + |396.81|)/8

Mean Deviation about the mean = (303.19 + 203.19 + 103.19 + 3.19 + 96.81 + 196.81 + 296.81 + 396.81)/8

Mean Deviation about the mean = 1403.19/8

Mean Deviation about the mean = 175.40

## Question:

Find the mean deviation about median for the following data: Marks Number of girls 0-10 6 10-20 8 20-30 14 30-40 16 40-50 4 50-60 2

## Answer:

Step 1: Find the median by arranging the data in ascending order: Marks Number of girls 0-10 6 10-20 8 20-30 14 30-40 16 40-50 4 50-60 2

Median = 20-30 (14 girls)

Step 2: Calculate the absolute deviation of each value from the median: Marks Number of girls Deviation from Median 0-10 6 -8 10-20 8 -6 20-30 14 0 30-40 16 6 40-50 4 -10 50-60 2 -12

Step 3: Add all the absolute deviations: -8 + -6 + 0 + 6 + -10 + -12 = -30

Step 4: Divide the sum of absolute deviations by the total number of values: -30/6 = -5

Therefore, the mean deviation about median is -5.