What is Statistics?
Statistics is a branch of mathematics that deals with the collection, organization, analysis and interpretation of data. In statistics one does not expect a single value as an answer to a question. Here we are interested to know the distribution and tendency of the answers.
What is Central Tendency?
Central tendency is a statistical measure that represents the entire dataset and gives it an accurate description. In other words, central tendency summarizes the dataset with a single value that is the center point or the typical value of a dataset.
There are three main measures of central tendency which are:
- Mean
- Median
- Mode
These measures indicate where most of the values in a distribution fall and are also referred to as the central location of a distribution. We can think of it as the tendency of data to cluster around a middle value.
Now let us discuss each of these in detail.
What is the Mean of a Dataset?
Mean of a data set is also called average. It can be calculated by dividing the sum of dataset values by the number of data values.
Mean gives us an overall idea of the values in a dataset.
\(Mean=\frac{sum~of~all~ data~values}{number~of~data~values}\)
Let us consider an example to understand the the mean of a dataset:
If the dataset values are 2, 3, 4, 5, 6, 7, 8, 9 then the mean of this dataset will be:
\(Mean=\frac{sum~of~data}{number~of~data}\)
If we count the number of values we will get 8, so the mean is
= \(\frac{2~+~3~+~4~+~5~+~6~+~7~+~8~+~9}{8}\) = 5.5
Therefore the mean of the given data set is 5.5.
What is Median of a Dataset?
If we arrange the values in a dataset in an increasing or ascending order, then the middle value is the median of the dataset.
Now to calculate the median we need to consider two cases:
- Number of dataset values is odd: In such a case the median will be the middle value.
- Number of dataset values is even: In this case the median will be the mean of the two middle values.
So if the number of dataset values is represented by n then the median is given by:
Median = value of data at \(\left(\frac{n~+~1}{2}\right)th~place\) [when n is an odd number]
Median = Mean of data at \(\left(\frac{n}{2}\right)th~place\) and \(\left[\left(\frac{n}{2}\right)~+~1\right]th~place\) [when n is an even number]
What is the Mode of a Dataset?
Mode of a dataset is that value in the dataset that occurs most often. A dataset can have one mode or more than one mode.
Let us look at an example:
Find the mode for the dataset: 1, 3, 6, 8, 4, 9, 4, 11
The value 4 occurs twice and the rest of the values occur only once. So the mode of the dataset is 4.
[Note: If all the values in a dataset occur the same number of times then we say that the dataset has no mode.]
Solved Central Tendency Examples
Example 1: Find the median of these soccer scores.
Solution:
5, 5, 6, 7, 8, 8, 9, 11 Order the scores in ascending order
Total number of scores mentioned = 8, which is an even number
For an even number of dataset values median is calculated with:
Median = Mean of data at \(\left(\frac{n}{2}\right)th~place\) and \(\left[\left(\frac{n}{2}\right)~+~1\right]th~place\)
= Mean of data at \(\left(\frac{8}{2}\right)th~place\) and \(\left[\left(\frac{8}{2}\right)~+~1\right]th~place\)
= Mean of data at 4th place and 5th place
= \(\frac{7~+~8}{2}\)
= \(\frac{15}{2}\)
= 7.5
Therefore the median score is 7.5.
Example 2: The table below displays the scores of various players in a match. Find the mean, median, and mode of the scores?
Name | Scores |
|---|---|
Thomas | 80 |
Andrew | 52 |
Tom | 40 |
Harris | 52 |
Karl | 70 |
Jerrico | 1 |
Kane | 6 |
Solution:
1, 6, 40, 52, 52, 70, 80 Order the dataset values in ascending order
1. Mean of dataset :
\(Mean=\frac{Sum~of~data}{Number~of~data}=\frac{1~+~6~+~40~+~52~+~52~+~70~+~80}{7}=43\)
Therefore, the mean score is 43.
2. Median of dataset:
Total number of dataset values is 7, which is an odd number.
Median = value of data at \(\frac{7~+~1}{2}th~place\)
= value of data at \(\frac{7~+~1}{2}th~place\) = 4th place
At 4th place we have 52
Therefore, the median score is 52.
3. Mode of Data set:
Since 52 repeats twice, therefore, the mode of the given data is 52.
The central tendency of a given dataset can be found using the formulas for mean, median and mode.
The significance of the central tendency is to provide an exact representation of the entire dataset. It is often defined as a single value that is representative of the complete dataset.
The mean represents the average of the given dataset. The value of the mean sometimes becomes equal to a given value of the dataset or it will differ from a value in the dataset.
When every value of a dataset is multiplied by a constant ‘k’, the mean of the dataset also becomes ‘k’ times the original mean value.