Post on 03-Dec-2014
description
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