Measures of Central tendency

Princy

Measures of Central Tendency or (Measures of Location)

  Enable us to locate the position of the

distribution in the question  Measures of central tendency are

measures of the location of the middle or the center of a distribution. The mean is the most commonly used measure

Summary Measures

Central Tendency

MeanMedian

Mode

Quartile

Geometric Mean

Summary Measures

Variation

Variance

Standard Deviation

Coefficient of Variation

Range

Harmonic Mean

Mean Deviation

Various measures of central tendency are

1. Arithmetic Mean2. Harmonic Mean Mathematical

Averages3. Geometric Mean4. Median 5. Mode Positional average

Measures of Central Tendency

Central Tendency

Mean Median Mode

Geometric Mean1

1

n

ii

N

ii

XX

n

X

N

1/

12

n

Gn XXXX

Harmonic Mean

Mean (Arithmetic Mean)

The Most Common Measure of Central Tendency

Affected by Extreme Values (Outliers)

(continued)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6

Mean (Arithmetic Mean)

Mean (Arithmetic Mean) of Data Values Sample mean

Population mean

1 1 2

n

ii n

XX X X

Xn n

1 1 2

N

ii N

XX X X

N N

Sample Size

Population Size

Arithmetic mean

The arithmetic mean is what is commonly called the average:

The mean of a numeric variable is calculated by

adding the values of all observations in a data set and then dividing that sum by the number of observations in the set. This provides the average value of all the data.

. It is a method representing whole data by one figure

.Example

Mount Rival hosts a soccer tournament each year. This season, in 10 games, the lead scorer for the home team scored 7, 5, 0, 7, 8, 5, 5, 4, 1 and 5 goals. What was the mean score?

Mean = sum of all the observed values ÷ number of observations           = (7 + 5 + 0 + 7 + 8 + 5 + 5 + 4 + 5 + 1) ÷ 10           = 47 ÷ 10           = 4.7//

 

Individual Series Example

The marks obtained by 11 students in Presidency College are

Roll no 1 2 3 4 5 6 7 8 9 1011

Marks 40 38 60 62 65 35 20 30 80 86 90 Calculate the Arithmetic mean?

ANSWER A.M x = x/N =

40+38+60+62+65+35+20+30+80+86+90 11 =

= 606 11

= 55.09//

Problem 2 The Mean weight of a student in a group

of 6 students is 119 lbs.The individual weight of five of them are 115,109,129,117,114 lbs.What is the weight of the sixth student?

130

Discrete series Discrete series refer to a series

when actual size of items(individual measurement) and corresponding frequencies are given

For finding the mean of a discrete variable three Methods

are there

Direct Method Short cut method Step deviation Method

Direct method The arithmetic mean is the sum of the

products of the values of the items and corresponding frequencies divided by total frequencies

Formula 

x = fx / f where f =N (total no of frequency)Where x is the sample mean, f is the sum of the frequencies in each class

Example 1

Calculate the arithmetic mean from the following data (by direct method)?

5 15 25 35 45 55 65 75 7 11 16 17 26 31 1 1

Answer

x f fx

5 7 3515 11 16525 16 400 Mean= 38.2735 17 59545 26 117055 31 170565 1 6575 1 75

110 4210

2.Short cut method

Formula  

Where A is the assumed mean , x is the actual mean, f or N is the sum of the frequencies in each

class

Example 1

Calculate the arithmetic mean from the following data by shortcut method

5 15 25 35 45 55 65 75 7 11 16 17 26 31 1 1

Answersx f d=x-A fd

5 7 -40 -280

15 11 -30 -330 Mean= A+f d/N

25 16 -20 -320

35 17 -10 -170 A=45

45 26 0 0

55 31 10 310 45+-740/110

65 1 20 20

75 1 30 30 38.27

110 -40 -740N fd

Arithmetic Mean by step Deviation Method

The Formula is

i is the magnitude of the class interval

Example 1

Calculate the arithmetic mean from the following data by step deviation method 5 15 25 35 45 55 65

75 7 11 16 17 26 31 1 1

Answerx f d=(x-A)/10 fd A=45

5 7 -4 -28

15 11 -3 -33 Mean= A+f d/N*I

25 16 -2 -32

35 17 -1 -17

45 26 0 0

55 31 1 31 45+(-740/110)*10= 38.27

65 1 2 2

75 1 3 3

110 -4.5 -74N fd

HW 1 Soccer tournament at Mount Rival I Mount Rival hosts a soccer tournament

each year. This season, in 10 games, the lead scorer for the home team scored 7, 5, 0, 7, 8, 5, 5, 4, 1 and 5 goals. What was the mean score?

Hw.2 The following table lists the number of people killed in traffic accidents over a 10 year period. During this time period, what was the average number of people killed per year? How many people died each day on average in traffic accidents during this time period?

Table 1. Number of fatalities in traffic accidents

Year Fatalities

1 959

2 1,037

3 960

4 797

5 663

6 652

7 560

8 619

9 623

10 583

HW 3In an office there are 84 employees. Their salaries are given

Below

Salary(Rs) 2430 2590 2870 3390 4720 5160Employees 4 28 31 16 3 2

1.Find the mean salary of the employees

2.What is the total salary of the employees

HW 4 The Calculate arithmetic mean of the

weight of ten students in a class

Sl No Wt1 422 563 494 505 496 537 528 489 47

10 54

HW 5 Find the missing frequency from the following

data

The average mark is 16.82 Ans: x = 18

Marks Frequency 0-5 10

5-10 12 10-15 16 15-20 ? 20-25 14 25-30 10 30-35 8

Hw 6 In the frequency distribution of 100 families given below the

number of families corresponding to the expenditure groups and 60-80 are missing from the table.However the mean of the distribution is known to be 50.Find the missing frequencies

Expenditure No of families0-20 14

20-40 -40-60 2760-80 -80-100 15

Thanks