Central Tendency: Definition, Methods and Sample Questions

The central tendency measure is a very convenient way of describing a set of scores with a single number that explains the performance of a group. In simple terms, the measure of central tendency is an average. It is a single set of values that can be considered typical in a collection of data.

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Definition of Measures of Central Tendency 

The Measures of Central tendency are numerical descriptive measures that indicate or locate the center of distribution or set of data. For instance, In a class of 50 students, the average height would be the average height of the class as a whole. Thus, It is also defined as a single value used to describe the center of data.

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Methods of Measuring Central Tendency

There are three types of central tendency measures that are commonly used. These are as follows:

1. Mean
2. Median
3. Mode

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What is Mean? 

• The sum of all measurements divided by the number of measurements in the set is the mean of a set of values or measurements.
• The mean is the most popular and widely used of the three measures of central tendency. It is also known as the arithmetic mean or arithmetic average.
• When we compute the population mean, we refer to it as the parametric or population mean. It is denoted by μ The symbol is read as "mu".

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What is Mean Value?

The mean value is the average value that falls between the maximum and minimum values in the data set, but it is not a number in the data set. A formula for Mean is as follows,

OR,

There is another method of calculating mean that is not widely used. This method is known as the Assumed Mean Method. In this case, a random value from the data set is chosen and assumed to be the mean. The deviation of the data points from this value is then computed. As a result, the mean is given by:

Mean = Assumed Mean + (Sum of All Deviations / Number of Data points)

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Example

Mean for Ungrouped Data

Miss Roma collects the data on the ages of History Teachers in St. Joseph School and her study gives the following data - 

Here, N = 7

So, Mean = 32+35+26+36+35+33+45 / 7 

= 245/7

= 35

Based on the above observation, 35 is the average age of History teachers in St. Joseph School.

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Weighted Mean

The weighted mean is the mean of a set of values with varying weights or degrees of importance. Its formula is as follows:

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Example

Vishakha subjects, as well as the corresponding number of units and grades for the previous grading period, are listed below. Make a note of her grade point average.

Thus, Solution = 1.5*90 + 1.5*86 + 1.8*88+0.9*87+1.5*87 / 1.5 + 1.5 + 1.8 + 0.9 + 1.5

= 135 + 129 + 158.4 + 78.3 + 130.5 / 7.2

= 631.2 / 7.2 = 87.67 is the general average of Vishakha.

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Properties of Mean

• Mean can be calculated or computed for any set of numerical data, it is always present.
• There is only one Mean in a set of numerical data.
• It takes into account every item in the data set, thus, the mean is the most reliable measure of central tendency.
• Extreme or deviant values have a significant impact on it.
• It is only used when the data is an interval or a ratio.

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What is Median?

When the data are ranked in order of size, the median, denoted Md, is the middle value of the sample.

The formula for calculating the data set's median is as follows:

Odd Number of Observation- If the total number of observations is odd, then the median is calculated as follows:

Median = {(n+1)/2}^thterm

where n = total number of observations

Even number of Observations- If the total number of observations is even, the median formula is as follows: 

Median = [(n/2)^thterm + {(n/2)+1}^th]/2

where n = total number of observations.

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Example 

Find the median of 16,17,18,19,20,21,22.

Solution: Let us put the given set of numbers in ascending order. So, it comes - 16,17,18,19,20,21,22

As the number of observations is odd,

Median = (N+1)/2 

= 7+1 / 2 

= 8/2 

= 4

The middle number is 19. Thus, the median is 19

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Properties of Median

• The median is the score or class in the distribution where 50% of the scores fall below it and 50% fall above it.
• Extreme or deviant values have no effect on the median.
• When there are extreme or deviant values, the median is appropriate to use.
• When the data is ordinal, the median is used.
• The median can be found in both quantitative and qualitative data.

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What is Mode?

The mode, denoted by Mn, is the most common value in a set of measurements or values. In other words, it is the most frequently encountered value in a given set.

Thus, the formula for mode is as follows:

Mode = l + (\(\frac{f_1 - f_0}{2f_1 - f_0 - f_1}\)) x h

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Example 

1. Determine the mode of the data set 7, 7, 7, 9, 19, 19, 19, 27, 37, 48.

Solution : The data set is as follows: 7, 7, 7, 9, 19, 19, 19, 27, 37, 48.

As we know, there can be more than one mode in a given data set or set of values. If more than one value occurs with equal frequency and number of times as the other values in the set. As a result, the numbers 7 and 19 are both modes of the set in this case.

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Properties of Mode

• It is used when you want to find the value that occurs the most frequently.
• It is a rough estimate of the average.
• It is the average of the inspections.
• Because its value is undefined in some observations, it is the least reliable of the three measures of central tendency.

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Points to Remember

• There are three types of central tendency measures that are commonly used. These are as follows: Mean, Median and Mode.
• Two modes can exist in a given set of data. Such values are referred to as bimodal.
• A measure of central tendency is a value around which other numbers gather.
• The median is the middle number when the given data is arranged in ascending order.
• The mode value is the least reliable of the three measures of central tendency.

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