MEASURES OF CENTRAL TENDENCY

Measures of Central Tendency

Measures of central tendency are a combination of two words i.e. ‘measure’ and ‘Central tendency’. Measure means methods and central tendency means average value of any statistical series. Measures of central tendency help us understand the general behavior or trend of a dataset, making it easier to draw conclusions and make decisions based on the data. Thus, we can say that central tendency means the methods of finding out the central value or average value of a statistical series of quantitative information.

Definition

1. J.P. Guilford has pointed out that “an average is a central value of a group of observations or individuals.”

2. According to Clark “Average is an attempt to find one single figure to describe whole of figure.”

3. In the words of A.E. Waugh “An average is a single value selected from a group of values to represent them in a same way—a value which is supposed to stand for whole group of which it is a part, as typical of all the values in the group.”

Thus it can be said that an average or central tendency is a single figure that is computed from a given distribution to give a central idea about the entire series. The value of the average lies within the maximum and minimum value in the series. There are three major ways to show central tendency: mean, mode and median.

How Measures of Central Tendency are useful in any research work

Measures of central tendency, such as the mean, median, and mode, are essential statistical tools in research for several reasons:

1. Summarizing Data: Central tendency measures provide a concise summary of a dataset by identifying a representative value around which other data points cluster. This simplifies the complexity of large datasets and makes them more understandable.

2. Identifying Representative Values: The mean, median, and mode help identify representative or typical values in a dataset, offering insights into the central or common characteristics of the observed phenomenon.

3. Data Comparison: Central tendency measures enable researchers to compare different groups or datasets. For example, comparing the average income in two regions provides a quick understanding of the economic conditions in those areas.

4. Decision-Making: Central tendency measures assist in decision-making by providing a point of reference. For instance, in business, the mean profit over several years can guide decisions on budgeting and resource allocation.

5. Understanding Variability: Alongside measures of dispersion, such as standard deviation, central tendency measures help in understanding the spread of data points around the central value. This is crucial for assessing the variability and reliability of the data.

6. Estimation: The mean is often used as a point estimator in inferential statistics to make predictions about a population based on a sample. It serves as an estimate of the population mean.

7. Data Cleaning: Identifying outliers or extreme values becomes easier with central tendency measures. Unusually high or low values may indicate errors in data collection and can be investigated further.

8. Statistical Tests: Many statistical tests and analyses, such as t-tests and analysis of variance (ANOVA), rely on measures of central tendency to assess differences and relationships between variables.

9. Communication: Central tendency measures offer a simple and clear way to communicate key characteristics of a dataset to a diverse audience, facilitating better understanding among stakeholders.

Central tendency refers to the measure used to determine the “center” of a distribution of data. It is used to identify a single value that represents an entire data set the most. The major types of central tendency are the mean, median, and mode.

Function of Measures of Central Tendency

The main functions of measures of central tendency are as follows:

1. They provide a summary figure with the help of which the central location of the whole data can be explained. When we compute an average of a certain group we get an idea about the whole data.

2. Large amount of data can be easily reduced to a single figure. Mean, median and mode can be computed for a large data and a single figure can be derived.

3. When mean is computed for a certain sample, it will help gauge the population mean. 4

4. The results obtained from computing measures of central tendency will help in making certain decisions. This holds true not only to decisions with regard to research but could have applications in varied areas like policy making, marketing and sales and so on.

5. Comparison can be carried out based on single figures computed with the help of measures of central tendency. For example, with regard to performance of students in mathematics test, the mean marks obtained by girls and the mean marks obtained by boys can be compared.

Measures of Central Tendency:

There are three measures of central tendency, such as:

(1) The Mean

(2) The Median

(3) The Mode

(1) The Mean (M):

The mean or the arithmetic average of a set of values of a variable is the value obtained after dividing the sum of all the values of the given variable by their number. For example, the mean of 2 3 5 9 11 is:

(2 + 3 + 5 + 9 + 11) / 5 = 30 / 5 = 6.

Summation Notation (∑):

The Symbol ∑ is the Greek capital letter sigma denoting sum. Let the symbol Xi (read x subscript i) denote any of the values X1 X2, X3… Xn, assumed by a variable x. The letter i stand for any of the number 1, 2…n is called a subscript.

Uses of Mean:

There are certain general rules for using mean. Some of these uses are as following:

1. Mean is the centre of gravity in the distribution and each score contributes to the determination of it when the spread of the scores are symmetrically around a central point.

2. Mean is more stable than the median and mode. So that when the measure of central tendency having the greatest stability is wanted mean is used.

3. Mean is used to calculate other statistics like S.D., coefficient of correlation, ANOVA, ANCOVA etc.

Merits of Mean:

1. Mean is rigidly defined so that there is no question of misunderstanding about its meaning and nature.

2. It is the most popular central tendency as it is easy to understand.

3. It is easy to calculate.

4. It includes all the scores of a distribution.

5. It is not affected by sampling so that the result is reliable.

6. Mean is capable of further algebraic treatment so that different other statistics like dispersion, correlation, skew-ness requires mean for calculation.

Demerits of Mean:

1. It is difficult to locate mean by mere inspection.

2. Sometimes, the result given by mean is difficult to follow.

3. Extreme items of the series affect the value of mean to a very large extent.

4. this gives satisfactory results when computed from homogenous extent.

5. It gives comparatively more importance to big items.

6. It often gives misleading conclusions.

7. Often it does not have value among the various item of the series.

(2) Median:

The median is determined by sorting the data set from lowest to highest values and taking the data point in the middle of the sequence. There is an equal number of points above and below the median.

For example, in the data set (1,2,3,4,5) the median is 3; there are two data points greater than this value and two data points less than this value.

Uses of Median: 

1. Median is used when the exact midpoint of the distribution is needed or the 50% point is wanted.

2. When extreme scores affect the mean at that time median is the best measure of central tendency.

3. Median is used when it is required that certain scores should affect the central tendency, but all that is known about them is that they are above or below the median.

4. Median is used when the classes are open ended or it is of un-equal cell size.

Merits of Median:  

1. It is easy to compute and understand.

2. All the observations are not required for its computation.

3. Extreme scores does not affect the median.

4. It can be determined from open ended series.

5. It can be determined from un-equal class intervals.

Demerits of Median:

1. It is not rigidly defined like mean because its value cannot be computed but located.

2. It does not include all the observations.

3. It cannot be further treated algebraically like mean.

4. It requires arrangement of the scores or class intervals in ascending or descending order.

5. Sometimes it produces a value which is not found in the series.

(3) Mode:

The mode of a set of numbers is that value which occurs reputedly with the greatest frequency and it is the most common value.

Let us consider the following figures:

3, 5, 8, 5, 4, 6, 5, 9, 5

Here, the mode of these numbers is 5 because it has appeared the highest time (4 times) in the series of these numbers.

Uses of Mode:

The mode is used:

1. When we want a quick and approximate measure of central tendency.

2. When we want a measure of central tendency which should be typical value. For example, when we want to know the typical dress style of Indian women i.e. the most popular dress style. Like this the average marks of a class are called modal marks.

Merits of Mode:

1. Mode gives the most representative value of a series.

2. Mode is not affected by any extreme scores like mean.

3. It can be determined from an open-ended class interval.

4. It helps in analyzing qualitative data.

5. Mode can also be determined graphically through histogram or frequency polygon.

6. Mode is easy to understand.

Demerits of Mode:

1. Mode is not defined rigidly like mean. In certain cases, it may come out with different results.

2. It does not include all the observations of a distribution but on the concentration of frequencies of the items.

3. Further algebraic treatment cannot be done with mode like mean.

4. In multimodal and bimodal cases it is difficult to determine.

5. Mode cannot be determined from unequal class intervals.

6. There are different methods and different formulae which yield different results of mode and so it is rightly remarked as the most ill defined average.