# Measures of Central Tendency and Dispersion

## Key Questions

#### Explanation:

Measures of central tendency are mean, mode and median. Even we have three types of mean, such as arithmatic mean, geometric mean and harmonic mean.

They tell us the central value around which the data is distributed. For example consider the data set $6 , 8 , 2 , 4 , 12 , 5 , 8 , 10 , 3 , 4$. In this sum of numbers is $62$ and as they are ten in number, mean is $\frac{62}{10} = 6.2$

Note that smallest number is $2$ and largest number is $12$. Now, even if we had set of numbers as $5 , 6 , 7 , 5 , 8$ and as sum of numbers is $31$ and they are five, mean is still $\frac{31}{5} = 6.2$. But $5 , 6 , 7 , 5 , 8$ are far more narrowly spread and hence nature of data is not very well brought out by just mean.

Similarly, we can have two data sets with same median or mode, but their spread may be different, as mode is just the more frequent among data points and median is the value of central data point, when the samme is arranged in increasing or decreasing order.

Measures of dispersion tell us better about the kind of spread. In a way, mean deviation or standard deviation tell us more about the way data is spread.

For example, data set $30 , 40 , 50 , 60 , 70$ and data set $10 , 30 , 50 , 70 , 90$ have same mean, mode and median but while mean deviation of first data set is $12$, that of second data set is $24$, indicating that second data set is too wide spread.

What about two data sets $30 , 40 , 50 , 60 , 70$ and $130 , 140 , 150 , 160 , 170$? Their mean deviation is same i.e. $12$, but are they not widely different as mean of first data set is $50$, while that of second data set is $150$.

It is obvious that measures of central tendency and measures of dispersion are both important and complementary.

• The standard deviation gives us an idea on how clustered together or how scattered our data is from the average.

If the standard deviation is small, we can say that the majority of our data are near the average.

If the standard deviation is large, we can say that our data are quite scattered.

• The measures of central tendency are the mean, mode, and median

The mean is the average of the given data.

The mode is the element among the data that occurs most frequently

The median is middle element when the data is sorted