Central tendency

Topics: Arithmetic mean, Mean, Average Pages: 20 (3485 words) Published: October 16, 2014
Introduction to Statistics (Measures of Central Tendency)

Central Tendency: In a representative sample, the value of a series of data have a tendency to cluster around a certain point usually at the center of the series is usually called central tendency and its numerical measures are called the measures of central tendency or measures of location.

Different Measures of Central Tendency: The following are the important measures of central tendency which are generally used in business:

Arithmetic mean
Geometric mean
Harmonic Mean
Median
Mode

Arithmetic mean: Arithmetic mean is defined as the sum of all observations divided by the total number of observations.

Calculation of Arithmetic Mean-Ungrouped Data: For ungrouped data, arithmetic mean may be computed by applying any of the following methods:

Direct method
Short-cut method

Direct method: The arithmetic mean, often simply referred to as mean, is the total of the values of a set of observations divided by their total number of observations. Thus, if represent the values of items or observations, the arithmetic mean denoted by is defined as: If the subscripts are dropped, the formula sample mean is: = and the population mean is:

Short-cut method: According to short-cut method arithmetic mean can be computed by the formula: , where , here A is called origin and h is called scale.

Example: The monthly expenditure (in taka) of 10 students given as follows:

14870149301502014460147501492015720151601468014890

Find monthly average expenditure.

Solution: Let income be denoted by X

By using calculator,
=149400
===14940
Hence, the average monthly income Tk.14940

Example: Mr. Peterson is studying the number of minutes used monthly by clients in a particular cell phone rate plan. Random sample of 12 clients showed the following number of minutes used last month.

90779489119112
911109210011383
What is the arithmetic mean number of minutes used?
Solution: Let the minute be denoted by X then =1170, === 97.5 The arithmetic mean number of minutes used last month by the sample of cell phone users is 97.5 minutes.

Calculation of Arithmetic Mean-Grouped Data: For grouped data, arithmetic mean may be computed by applying any of the following methods:

Direct method
Short-cut method

Direct Method: When direct method is used
=
Where, = mid-point of the different classes
= the frequency of each class
= the total frequency ()

Note: For computing mean in the case of grouped data the mid points of the various classes are taken as representative of that particular class. The reason is that when the data are grouped, the exact frequency with which each of the variable occurs in the distribution is unknown.

Example: The following are the figures of profits earned by 1400 companies during 1999-2000.

Profits (Tk. lakhs)
No. of companies
Profits (Tk. lakhs)
No. of companies
200-400
500
1000-1200
100
400-600
300
1200-1400
80
600-800
280
1400-1600
20
800-1000
120

Calculate the average profits for all companies.

Solution:

Calculation of average profits

Profits (Tk. lakhs)
Mid-point
No. of companies

200-400
300
500
150000
400-600
500
300
150000
600-800
700
280
196000
800-1000
900
120
108000
1000-1200
1100
100
110000
1200-1400
1300
80
104000
1400-1600
1500
20
30000

We know that, = then using data from table, = = 605.71
So the average profit is 605.71 lakhs taka.

Arithmetic mean for two or more related groups: If we have the arithmetic mean and number of observations two or more than two related groups, we can compute combined average of these groups by applying the following formula.

=
Where,
=Combined mean of the two groups, =Arithmetic mean of the first group =Arithmetic mean of the second group, =No. of observations in the first group = No. of observations in the second group

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