The word central tendency indicates the middle Value and is measured using the mean median mode. Each of these measures are calculated differently and all these are measured in different situations depending upon the occurence of the date. Its the degree of clustering of values of the distribution and is calculated using the above measures.
Central Tendency- Mean
The most commonly used measure of a central tendency is the mean. It is also called the average. A mean deviation is defined as the sum of the items of a data divided by the number of items in the data.
For example, consider a data of set of numbers 2, 5, 6, 10, 12. The sum of the items is the sum of the numbers 2 + 5 + 6 + 10 + 12 = 35 and number of items is 5. Hence the mean of this data is, $\frac{35}{5}$ = 7.
Central Tendency- Median
The median of a data is the middle item of the data when all the items are arranged in an order.
For example, consider a data of set of numbers 2, 8, 6, 12, 10.
The items of the data when arranged in order (say from least to greatest) are,
2, 6 , 8, 10, 12
The middle item is 8 and hence 8 is the median of this data.
In case of a data with even number of items, the mean of the middle two terms is the median of the data.
Median is a better measure of central tendency compared to mean if the data is skewed and also it is not influenced by the presence of outliers. We will see this in the illustrated example.
Central Tendency- Mode
The mode of a data is the item or items which occur the most out of all the items of the data, when all the items are arranged in an order. There may not be any mode in some data or there may be more than one mode in some data.
For example, consider a data of set of numbers 2, 8, 6, 12, 10.
The items of the data when arranged in order (say from least to greatest) are,
2, 6 , 8, 10, 12
All the items in the data occur only once. Hence there is no mode or nil mode for this data.
But consider a data of set of numbers 2, 8, 6, 2, 6, 12, 10, 6, 8, 3, 8
The items of the data when arranged in order (say from least to greatest) are,
2, 2, 3, 6, 6, 6, 8, 8, 8, 10, 12
In this data, the number 2 occurs twice, the number 6 occurs thrice and also the number 8 occurs thrice. Hence there are two modes for this data which are 6 and 8.
When the data represents a category, the mode of the data tells us the most favorite item.
Central Tendency Examples
Find the mean, median and mode of the set 82, 89, 83. 81, 82, 10
The items of the data when arranged in order (say from least to greatest) are,
10, 81, 82, 82, 83, 89
The mean of the data = 10 + 81 + 82 + 82 + 82 + 89 = $\frac{426}{6}$ = 71
There are two middle terms 82 and 82 and hence the median is $\frac{(82 + 83)}{2}$ = 82.5
The number 82 occurs most and hence the mode is 82.
The number 10 is far different from other numbers in the set. Such an item is called an outlier in the set. Suppose we ignore the outlier and calculate the mean, it becomes as, 81 + 82 + 82 + 82 + 89 = $\frac{416}{5}$ = 83.20. It can be seen now that an outlier has a great influence on the mean but has little influence on median or mode. Hence a more realistic conclusion of a central tendency is the measure of median.
Suppose the given data is the set of the scores by a student. The median gives a better report about the student than the mean. The score of 10 may be incorrectly awarded or the student might have taken that particular test in a hard situation. But that score drastically reduces his mean. Normally a good judge will go by the median in such cases.
Suppose the numbers are the code numbers of different commodities sold by a shop. The mode 82 tells that the commodity referred by the code 82 is more popular than the rest of the commodities.