# 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

*** If we have...

Please join StudyMode to read the full document