31 Measures of Central Tendency

1.  Introduction

1.   What is measure of central value?

A measure of central tendency gives a single representation value for a set of data. The single value is the point of location around which the individual values of the set cluster. The measures of central tendency are hence known as ‘measures of location’. They are popularly called averages.

Various measures of central values are the following:

• Arithmetic Mean
• Median
• Mode
• Quartiles
• Geometric Mean
• Harmonic Mean

    2.  Objectives: This module covers the following aspects of averages

1. Objectives or Functions of an Average
2. Criteria or characteristics or desirable properties of an average
3. Arithmetic Mean
4. Median
5. Mode
6. Objectives of an Average

  First, average provides a quick understanding of complex data:

• Secondly averages enable comparison:That istwo or more sets of values can be compared on the basis of their averages.
• Thirdly, average facilitate sampling techniques:That is the average of a sample tells what is the average of the population likely to be.
• Fourthly, averages pave the way for further statistical analysis
• Fifthly, averages establish the relationship between variables:That isthe statistical techniques such as correlation and regression are used to assess the relationship between variables. The above techniques are based on averages.

4. Criteria or properties of an average Next let us discuss criteria or characteristics or desirable properties of an average:

• Firstly, averages must be rigidly defined.That is It should have a well defined formula and procedure.
• Secondly, it should be based on all the items.
• Thirdly, it should not be unduly affected by extreme items.
• Fourthly, it should lend itself for algebraic manipulation.
• It should be simple to understand and easy to calculate.
• It should have sampling stability. It means that the results must be same for all samples.

5.  Arithmetic Mean What is arithmetic mean?

Arithmetic Mean is the total of the values divided by the number of observation.

Methods of Finding Arithmetic Mean:

Mean can be calculated by

(i) Direct Method

(ii) Shortcut Method and

(iii)Step Deviation Method.

All the methods give the same result for a problem. All the methods are illustrated.

Data – Type I (Individual Observations or Raw Data)

When the observed values are given individually such asX1, X2, X3, …. XN the methods of calculation of arithmetic mean are as follows:

Direct Method:

Total of the observations Arithmetic Mean= Number of the observations = SX /N

The calculation consists of the following two steps:

Step 1: Denote the given observations by X and find their total, SX.

Step 2: Identify N. the number of observations and divide SX by N.

For example,the expenditure of 10 families in Rupees are given below:

Short cut Method. Arithmetic mean may also be calculated through short cut method. If the value is very high, we can simplify the calculation. The formula in the short cut method is X̅ = S  / Where

Sd is the total of the deviations, that is sum of X-A, A is the assumed mean values which may be from X or may not be one of the given X values.

The following four steps are involved in the calculation. Step 1: Choose certain value for A if it is not specified in a problem

Step 2: Find the deviations of X from A. That is, calculate d=X-A corresponding to each X.

Step 3: Find Sd.

Step 4: Identify N and find  X = A + S   /N

For the above same example given in the direct method, calculate arithmetic mean by using short cut method

Let A = 100, N = 10, Sd = 61.Substitute all the values in the formula        X̅ =   + S /N.

That is 100+61/10 =106.10.

Data – Type II (Discrete Series)

If the actual values with corresponding frequencies are given,the method of calculation of arithmetic mean are as under:

Direct Method: Arithmetic mean = S

The calculation consists of the following four steps:

Step 1: Form a table with columns X and f.

Step 2:FindfX.

Step 3: Find N (=Sf) and calculateSfX

Step 4: Divide SfX by N to get the value of X̅.

Based on the above steps, the arithmetic is calculated for the data given below.

Calculate the mean number of persons per house.

The following five steps are involved in the calculation

Step 1: Form a table with columns X and f.

Step 2: Choose certain value for A if it is not specified in a problem. Form the column with the values of d (=X-A).

Step 3: Form the next column with title fd. Multiply the values of f and d in pairs and enter the products in that column.

Based on the above steps, we can calculate arithmetic mean for the above example in the direct method.

The same answer is obtained as 4.35 as in the direct method.

Data – Type III (Continuous Series – Exclusive Class Intervals)

Direct Method

In the continuous series, mid value is calculated as the range of value is given.

For example Calculate the arithmetic mean for the following.

Marks	20-30 30-40	40-5050-60    60-7070-80
No. of Students   5   8	12   15	6	4

Solution:

Calculate mid values as 20+30/2 = 25 and so on

Find fm values that is fm = 24650 and n= 50

Substitute the above values in the formula

Arithmetic Mean – continuous series – Shortcut method

The following are the steps to calculate arithmetic mean using shortcut method

Step 1: Form a table with X values and f.

Step 2: Form the mid values (m)m =  +    2

Step 3: Choose A and find d = m-A corresponding to each m.

Step 4: Form fd That is, products of f and d pair wise.

Step 5: Find N (=Sf) and Sfd.

Step 6: Find – from – =a+(sd)

Based on the above steps, calculate arithmetic mean for the data given in the direct method. The same answer is derived in the short cut method. That is 49.20.

Step Deviation Method: The steps under step deviation method are as follows:

Step 1: Form a table with X and f values.

Step 2: Form the mid values (m) in the next column m =    + 2

Step 3: Choose A = assumed mean  and C = class interval. Find the step deviations d’ =      −

Step 4: Find the products of f and d’ pair wise and enter them in the next column.

Step 5: Find N and Sfd’. S d’

Step 6: Find X by using the formula X = A + (  ) The same answer is obtained as 49.20 for the same data in the direct method.

6.MEDIAN

What is median?

Median is the value of the middle most item in the given data set when the data is in ascending or descending order.

Median is A POSITIONAL AVERAGE as it is calculated based on its location and its position in the data series.

Median divides the series into two equal parts. Half of the items will be equal to or more than the median.

Methods of Finding Median

For individual observations

Median = Size of N+1/2th item

For example calculate median for the following data which is in odd number 1   3       2   4       569810

First arrange the data in ascending order 1 234568910

Secondly apply the formula Size of N+1/2th item

That is 9+1/2 =5th item

5th item in the series is 5 which is median

Discrete Series:

In a discrete series, each value occurs corresponding frequency times. No value is written repeatedly. ‘f’ indicates how many times it occurs. For example if f = 5 for X = 2, 2 is not written 5 times. Hence, for finding the value at the middle most position, values are considered in ascending order and cumulative frequencies are formed.

frequency of 8 in the table. It is 2. Hence, Median = 2.When   +12is a fraction, the two middle most items are to be identified in a similar manner and the mean of those two items is to be found.

Continuous Series:

In individual observations and discrete series, the actual values of X are known. Hence, median is found out as seen earlier. In continuous series, the actual values of X are not known. Hence, another procedure is followed. That is first median class is identified. After that additional one more formula is used to identify the median value in the specified median class.

The following formula is used to locate median such that 2 items are less than Median and 2 items are more than Median: M = L + ( (  /2−    )) xi

The class interval which contains 2th cumulative frequency is the median class interval or the median class. That is, median lies within that class interval.

L – Lower limit of the class , f is the frequency of that class,

I is the class interval (i= lower limit –upper limit of a class) and cf is the cumulative frequency of the class preceding the median class.

The calculation of median from a continuous series has the following steps:

Step 1: Consider the data such that the class intervals are continuous and are in ascending order. The data are to be revised, if necessary.

Step2: Find(less than) cumulative frequencie

Step 3: Calculate 2 and identify the median class interval.

Step 4: Identify the values of L, f, I and cf.

Step 5: Substitute in the formula and simplify.

Step 6: Ascertain that the value of M lies in the median class interval. Otherwise, check the steps. Based on the above steps, median is calculated as under.

7MODE

What is mode?

Mode is the value which has the greatest frequency density.

i.e., it is the most common value. The mode may not exist and even if it does exist it may not be unique”. There is no mode when all the observations occur equal number of times. If one value occurs more times than any other value, that value is the mode. The set which has only one mode is said to be unimodal. Sets with two modes are said to be bimodal. Sets which have more than two modes are said to be multimodal. In a few situations due to fluctuations of sampling it becomes a difficult task to identify a single value with greatest frequency in a sample even though the population is undoubtedly unimodal.

Calculation of mode

Individual observations:

Example: Determine the mode:

(i) 320, 395, 342, 444, 551, 395, 425, 417, 395, 401, 390, 400.

Solution:

(i) Mode = 395 because its frequency 3, is higher than others. The frequency of others is 1 each (This is an example for unimodal distribution).

Discrete Series:

Identify the value which has the greatest frequency as mode. If the difference between the greatest frequency and the next lower frequency is not much, form the grouping table and analysis table and find the value which has the greatest frequency density as the mode.

1. Take and write the X and f values in the first two columns
2. Add two frequencies first and write in separate column
3. Leave first frequency, add two frequencies and form 4thcolumns.
4. Add three frequencies and form 5thcolumn
5. Leave first frequency, add three frequencies and form 6th column
6. Leave first two frequencies, add three values and form 7thcolomn

  To form analysis table, find out the highest frequency in each frequency column and identify corresponding X values and give value one for each .

In the original frequency column, the highest value is 20 which belongs to X value 14, in the second column, the highest value of frequency is 34 which belongs to X values of 12 and 13. For the above stated x values, give values 1 for each. Like that find values for each frequency column and find out the highest frequency x value. That is mode.

The highest value is obtained for the X value of 13 Hence Mode = 13.

8. CONCLUSION

Let us summaraise, various measures of central values are calculated with examples. It may give some idea to calculate various measures of central values. The use of various types of central value depends on the purpose of calculation and type of data. Each measure of central value has its own merits and limitation. For example, arithmetic mean is affected by extreme values. While the median is not based on all the items as it is a positional average. In this module, geometric mean, harmonic mean, quartiles deciles are not included due to the scope of the module though they are the measure of central values. Try to calculate measures of central values for some more examples from text books which will give you more practice.