Measures of Central Tendency

Measures of Central Tendency
Objectives of the chapter
• To use summary statistics to describe collections of data • The main goal is to come up with the one single number that best describes a distribution of scores.
• Lets us know if the distribution of scores tends to be composed of high scores or low scores.
• To use the mean, median and mode to describe how data bunch up.

The sales of 100 fast food shop is given below:
Sales
No. of
(in 000s) Shops
700-799
4
800-899
7
900-999
8
1000-1099 10
1100-1199 12
1200-1299 17

Sales
(in 000s)
1300-1399
1400-1499
1500-1599
1600-1699
1700-1799
1800-1899

No. of shops 13
10
9
7
2
1

Measure of Central Tendency
The following are the important measures of
Central Tendency
a)Mean b) Median and c) Mode
Mean can be classified as three types:
i) Arithmetic mean ii) Geometric mean iii) Harmonic mean.

Measure of Central Tendency
 An average (mean) is a single value which is considered as the most representative or typical value for a given set of data.
 Such a value is neither the smallest or the largest value, but is a number whose value is somewhere in the middle of the group.
 Measure of Central tendency show the tendency of some central value around which data tends to cluster.

Measure of Central Tendency
Objectives of averaging:
1.To get one single value that describes the characteristics of the entire data.
2.To facilitate comparison
Characteristics of good average:
1.It should be easy to understand.
2.It should be simple to compute
3.It should be based on all the observation.

Measure of Central Tendency
Characteristics of good average:
4. It should be rigidly defined.
5. It should be capable of further algebraic treatment 6. It should have sampling stability.
7. It should not be unduly affected of extreme values. Measure of Central Tendency
Arithmetic mean: The most popular and widely used measure for representing the entire data by one value is called arithmetic mean.
• It is calculated by adding together all the observations and dividing this total by the
X
number of observations.
X = n Calculation of arithmetic for Ungrouped Data:
1.Direct method
2.Short-cut method.

Measures of Central Tendency
Mean
• The arithmetic average, computed simply by adding together all scores and dividing by the number of scores.
• It uses information from every single score.
• For a population:

X
=
N

X
For a Sample: X = n Measures of Central Tendency

Define the population mean by the formula for group data μ = Σfi xi/N

where

μ = the population mean
Σxi = the sum over each member of the population xi fi= is the frequency of that class
N = the number of items in the population

Measures of Central Tendency
For the 60 temperature readings in this population we obtain: 87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,
73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,
84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,
85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68 μ = (87+85+ 79 +….+72+68)/60 = 4751/60 = 79.183

Measures of Central Tendency
In the previous pages we have calculated the mean from the raw data. We can also use the tabulated data to calculate the mean of the population
Use the formula

μ = Σ(fi * xi) / Σ fi

Where

xi = the midpoint of the ith class and fi = the number of items in the ith class

Measures of Central Tendency
From the table we obtain
Class

Class Midpoint (x) Frequency (f)

f*x

65- 69

67

6

402

70 – 74

72

11

792

75 – 79

77

20

1540

80 – 84

82

13

1066

85 – 89

87

9

90– 94

92

1
60

783
92
4675

Mean=Σ(fi*xi)/N = 4675/60 = 77.92
The small discrepancy between these two values for the mean is due to the way the data is accumulated into classes. The mean of the raw data is more accurate, the mean of the tabulated data is often more convenient to obtain.

Properties of Mean:
1.The algebraic sum of the deviations of the observations from their mean is zero.
2.If a and b are constants such that x=a+by where x and y are variables. Then

x a  by
3. If u=x+y then u y are variables.

x  y

where u, x and

Correcting Incorrect Values:
1.The average weekly wage for a group of 25 persons working in a factory was calculated to be 300. It was later discovered that one figure was misread as 500 instead of the correct value 1000. Calculate the correct average.
2.The mean of 200 observations was 50. Later on, it was discovered that two observations were wrongly read as 92 and 8 instead of
192 and 88. Find out correct mean.

Median
The median is the measure of central tendency which appears in the middle of an ordered sequence of values . That is half of the observations in a set of data are lower than it and half of the data are upper than it.

For example: Production cost of some fans are Tk.2000, 2300, 1200, 3500, 1100.
Arrange the data in ascending order as
1100, 1200, 2000, 2300, 3500.
Since there is 5 observations .The middle value is the 3rd number which is 2000
Then median of the above data is Tk.2000

Median
If N is odd number then Median=(N+1)/2 th observation of the data.
If N is even number then Median=is the average of N/2th and (N+1)/2th number.
 Find median of the data:
456, 544, 678, 892, 256, 345

Median for Grouped Data
Formula for calculation:
Median
h

n
Me L0  (  Fc) f 2

Here L0= lower limit of the median Class h= class interval of that class f= frequency of the median class n= No. of observation
Fc= Cumulative frequency of the pre-median class The sales of 100 fast food shop is given below. Find Median value of the sales
Sales
No. of
(in 000s) Shops
700-799
4
800-899
7
900-999
8
1000-1099 10
1100-1199 12
1200-1299 17

Sales
(in 000s)
1300-1399
1400-1499
1500-1599
1600-1699
1700-1799
1800-1899

No. of shops 13
10
9
7
2
1

The Mode
• The mode (or modal value) of a variable in a set of data is the value of the variable that is observed most frequently in that data or the value that is repeated most often in the data.
– Note: the mode is the value that is observed most frequently, not the frequency itself. (I see this error too frequently on tests.)
• The mode is defined for every type of variable [i.e., nominal, ordinal, interval, or ratio].
– However, the mode is used as a measure of central tendency primarily for nominal variables only.

The Mode (cont.)
• The mode is defined for ordinal and interval variables as well.
– The mode may be ill-defined if we have either: • a small number of cases; or
• a precisely measured continuous variable and a finite number of cases;
– because in either event it is likely that no value will be observed more than once in the data.

×h

How To Calculate the Mode
For Group data:
Mode:

The sales of 100 fast food shop is given below. Find Modal value of the sales
Sales
No. of
(in 000s) Shops
700-799
4
800-899
7
900-999
8
1000-1099 10
1100-1199 12
1200-1299 17

Sales
(in 000s)
1300-1399
1400-1499
1500-1599
1600-1699
1700-1799
1800-1899

No. of shops 13
10
9
7
2
1