Central Tendency

Measures of Central Tendency:

The central tendency of any data set can be measured by anyone of the following ways.

• Mean

• Median

• Mode

In general, the mean is the average of the data in the dataset. Median is the midpoint of the set of grouped or ungrouped data. Mode is the value that is repeated a maximum number of times in the set of data. 

Central Tendency in Statistics:

According to experts, a good measure of central tendency in statistics should have the following features.

• Measures of Central tendency definition should be easy and rigid. The calculations should be simpler. 

• The calculation should give one particular value. There should be no ambiguity as to the value of the central tendency obtained. 

• The measures of central tendency should represent the entire set of data. To be more specific, the value should lie between the upper limit and the lower limit of the set of data. 

• The measured value of central tendency should be flexible enough to fit into the further statistical calculations. 

• The measures of central tendency should not alter with the alteration is the extreme values of the data set. 

Measures of Central Tendency Definition:

There are three measures of central tendency namely mean, median, and mode.

Mean:

Mean is the measure of the average of a set of values which can be calculated by dividing the sum of all the observations by the number of observations. There are four different ways to measure the mean of a data set. They are arithmetic mean, geometric mean, harmonic mean, and weighted arithmetic mean. Usually, arithmetic mean is calculated because it is easy to calculate. 

Median:

Median is the middle value of the given data set. Usually, the data is arranged in ascending order before determining the median. If there are an odd number of observations, the median is the value represented by the middle of the data set. However, when there are an even number of observations, the median will be calculated as the mean of the two middle values. 

Mode:

Mode is the observation in the data set which occurs most frequently. In other words, the value that has repeated more times in a data set is called its mode. Mode can sometimes have multiple values. In certain cases where all the values are unique, the mode does not exist at all. 

Relationship Between Mean,Median, and Mode of a Data Set:

The relation between the values of mean, median, and mode of a given data set is established by empirical formula. Empirical formula states that the mode of a distribution is equal to the subtraction of 2 times its mean from 3 times its median.

Central Tendency Measures Example:

1. Consider the scores of 10 students in class X in a mathematics examination as 45, 30, 75, 90. 64, 72, 85, 62, 45, 97. Calculate the mean and median of the data set and also find the mode using an empirical formula. 

Solution:

The given data set should be arranged in ascending order.

30, 45, 46, 62, 64, 72, 75, 85, 90, 97

The mean of the given data is calculated by adding all the numbers and dividing it by 10.

\[Mean = \frac{{30 + 45 + 46 + 62 + 64 + 72 + 75 + 85 + 90 + 97}}{{10}} = \frac{{666}}{{10}} = 66.6\]

Median is calculated as the mean of 2 middle values because the number of values is 10 which is an even number. 

\[Median = \frac{{64 + 72}}{2} = \frac{{136}}{{10}} = 68\]

Empirical formula is given as:

\[Mode = 3 \times Median - 2 \times Mean\]

Mode = 3 (68) - 2 (66.6)

Mode = 204 - 133

Mode = 70.8

Therefore, the mean, median and mode of the given data set are 66.6, 68 and 70.8

Fun Facts about Central Tendency Definition:

• Mean is usually used to measure the central tendency of a symmetric data set. It is not used to measure the central value of skewed continuous data set because, in this type of data, the mean is pulled away from the center due to extreme values. 

• The choice of central tendency which is to be measured for a given set of data is based on the type of data.