# Measures of Central Tendency

**Mean.**

(i) Mean (for ungrouped data) = where x_{1}, x_{2}, x_{3}, ..., x_{n}are the observations and n is the total no. of observations.

(ii) Mean (for grouped data) = , where x_{1}, x_{2}, x_{3}, ..., x_{n}are different variates with frequencies f_{1}, f_{2}, f_{3}, ..., f_{n}respectively.

(iii) Mean for continuous distribution.

Let there be n continuous classes, y_{i}be the class mark and f_{i}be the frequency of the ith class, then

mean = (Direct method)

Let A be the assumed mean, then

mean = A +, where d_{i}= y_{i}-A (Short cut method)

If the classes are of equal size, say c, then

mean = A +c x , where u_{i}= (Step deviation method)**Median.**

(i) Median is the central value (or middle observation) of a statistical data if it is arranged in ascending or descending order.

(ii) Let n be the total number of observations, then

Median = **Quartiles**

(i) Lower Quartile =

(ii) Upper Quartile =

(iii) Inter quartile-range = upper quartile -lower quartile**Mode.**

(i) Mode (or modal value) of a statistical data is the variate which has the maximum frequency.

(ii) The class with maximum frequency is called the modal-class.

## Exercise

- Calculate the arithmetic mean of 5·7, 6·6, 7·2, 9·3, 6·2.
- The marks obtained by 12 students in a class test are 14, 13, 09, 19, 05,
08, 16, 17, 11, 10, 12, 16. Find

(i) the mean of their marks.

(ii) the mean of their marks when the marks of each student are increased by 3.

(iii) the mean of their marks when the marks of each student are doubled. - The mean of the numbers 6, x, 7, 14, 3x +3 is 10, find the value of x.
- Find the mean of 25 given numbers when the mean of 10 of them is 13 and the mean of the remaining numbers is 18.
- Find the median of the following numbers:

(i) 53, 50, 48, 51, 78, 86, 45

(ii) 3, 5, 1, 2, 4, 6, 0, 2, 2, 3

(iii) 9, 0, 2, 8, 5, 3, 5, 4, 1, 5, 2, 7 - If 3, 8, 10, x, 14, 16, 18, 20 are in ascending order and their median is 13, find the value of x.
- Calculate the mean and the median of the numbers 2, 1, 0, 3, 1, 2, 3, 4, 3, 5
- Calculate the mean and the median of the following set of numbers:

1, 9, 10, 8, 2, 4, 4, 3, 9, 1, 5, 6, 2, 4 - The mean of the numbers 1, 7, 5, 3, 4, 4 is m. The numbers 3, 2, 4, 2, 3,
3, p have mean m -1 and median q. Find

(i) p

(ii) q

(iii) the mean of p and q. - Find the mode of the following numbers:

(i) 3, 5, 1, 2, 4, 6, 0, 2, 2, 3

(ii) 9, 0, 2, 8, 5, 3, 5, 4, 1, 5, 2, 7 - Calculate the mean, the median and the mode of the numbers 3, 2, 6, 3, 3, 1, 1, 2
- Find the median, lower quartile, upper quartile and the semi-interquartile range of the following numbers 13, 17, 20, 5, 3, 19, 7, 6, 11, 15, 17
- The marks obtained by 16 students in a class test are:

3, 6, 8, 13, 15, 5, 23, 21, 9, 10, 17, 20, 1, 18, 12, 21

Find: (i) the median (ii) lower quartile (iii) upper quartile. - In a class test, the marks (out of 10) of 30 students were 5, 3, 6, 5, 5,
4, 1, 3, 8, 4, 3, 6, 5, 4, 8, 2, 5, 4, 4, 4, 3, 7, 2, 4, 5, 4, 8, 9, 10, 7.
Draw a tally chart and a frequency table. Hence find

(i) the mean (ii) the median (iii) the mode. - Find the median and the mode for the following distribution:

Wages per day (in rupees) 38 45 48 55 62 65 No. of workers 14 8 7 10 6 2 - Marks obtained by 70 students are given below:

Marks 20 70 50 60 75 90 40 No. of students 8 12 18 6 9 5 12

Find: (i) the median marks, (ii) the modal marks.

[**Hint.**Arrange the variates in ascending order.] - The distribution of heights of 50 students (measured to the nearest cm) was as under:

Height 110 115 118 120 121 125 No. of students 6 8 14 15 4 3 - Calculate the mean height for this distribution correct to one place of decimal.
Category A B C D E F G Wage in Rs/day 50 60 70 80 90 100 100 No. of workers 2 4 8 12 10 6 8

(i) Calculate the mean wage, correct to the nearest rupee.

(ii) If the number of workers in each category is doubled, what would be the new mean wage? - Find the value of p for the following distribution whose mean is 20·6

Variate(x _{i})10 15 20 25 35 Frequency(f _{i})3 10 p 7 5 - For the following frequency distribution, find

(i) the median (ii) lower quartile (iii) upper quartile.

Variate 25 31 34 40 45 48 50 60 Frequency 3 8 10 15 10 9 6 2 - Calculate the mean, the median and the mode of the following distribution:

No. of goals 0 1 2 3 4 5 No. of Matches 2 4 7 6 8 3 - Calculate the mean, the median and the mode of the following distribution:

Age in years 12 13 14 15 16 17 18 No. of students 2 3 5 6 4 3 2 - Calculate the mean, correct to one decimal place for the following distribution:

Marks 0-10 10-20 20-30 30-40 40-50 No. of students 2 5 20 8 7 - Following table gives marks scored by students in an examination:

Marks 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 No. of students 3 7 15 24 16 18 5 2

Calculate the mean mark correct to 2 decimal places. - Weights of 50 eggs were recorded as given below:

Weight in gms 80-84 85-89 90-94 95-99 100-104 105-109 110-114 No. of eggs 5 10 12 8 2 1

Calculate their mean weight to the nearest gram.

## Answers

**1.**7

**2.**(i) 12·5 (ii) 15·5 (iii) 25

**3.**5

**4.**16

**5.**(i) 51 (ii) 2·5 (iii) 4·5

**6.**12

**7.**Mean = 2·4, median = 2·5

**8.**Mean = , median = 4

**9.**(i) 4 (ii) 3 (iii) 3·5

**10.**(i) 2 (ii) 5

**11.**Mean = 2·625, median = 2·5, mode = 3

**12.**13, 6, 17, 5·

**13.**(i) 12·5 (ii) 6 (iii) 18

**14.**(i) 4·93 (ii) 4·5 (iii) 4

**15.**Median = Rs 48; mode = Rs 38

**16.**(i) 50 (ii) 50

**17.**117·8 cm

**18.**(i) Rs 85 (ii) Rs 85

**19.**25

**20.**(i) 40 (ii) 34 (iii) 48

**21.**Mean = 2·77, median = 3, mode = 4

**22.**Mean = 14·96, median = 15, mode = 15

**23.**28·1

**24.**18·56

**25.**94 gm