Measures of Central Tendency- Mean, Median and Mode

A measure of central tendency is a summary measure to describe a whole data set with a single value, representing the middle or centre of its distribution.

There are three measures of central tendency to represent numerical data: the mean, median and mode

Mean

The mean of a data set is an average score.

Suppose there are three friends planning a trip to Sydney. They plan to fly there but the night before, they find that there is a weight limit of 20 kg on their luggage per person. They weigh their luggage and find their luggage weighs 16 kg,18 kg,23 kg forming a numerical data set of 16, 18, 23

Since one of them has packed too much, they decide to share their luggage around so that they all carry the same amount. How much does each person need to carry now? In a more mathematical language, we can say that they are sharing the total luggage equally among the three groups. This can be written as a mathematical expression 

$$\frac{17+18+22}{3}=\frac{57}{3}=19$$

Each person carries $19\text{ kg}$. This amount is the mean of the data set.

If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same.$$\text{<span style="font-family: arial;">Mean</span>}<span\; style="font-family:\; arial;">=</span>\frac{\text{<span style="font-family: arial;">Sum of scores</span>}}{\text{<span style="font-family: arial;">Number of scores</span>}}$$

Median

The median of a data set is another average.

Suppose there are 9 people in a room. Their weekly income responses form the data set

$200,$300, $400, $400, $430, $450, $470, $500, $2900

If we find the mean of this data set we get 

200+300+400+400+430+450+470+500+2900 = 820

                               9

This mean amount does not represent the data set well as 8 out of 9 people earn much less than this. 

In such cases, finding the median which is the middle score of a data set represents the data set.

To find the median, a data set is arranged from the smallest to the largest scores.

Instead, we can select the median, the "middle" score. We remove the biggest and the smallest scores to get:$300, $400, $400, $430, $450, $470, $500

Then the next biggest and the next smallest to get:$400, $400, $430, $450, $470

Then the next biggest and the next smallest to get:$400, $430, $450

Then the next biggest and the next smallest to get: $430

The only number left is the median for this data set which is $430. This median weekly income represents and summarises the set better. 

Therefore, the median is the middle score when there is an odd number of scores. However, when there is an even number of scores, the median will be the number between the two scores.

Half of the scores are greater than the median and half are lesser than the median.

From https://mathspace.co/textbooks/syllabuses/Syllabus-842/topics/Topic-18502/subtopics/Subtopic-251071/?selectTextbook=false#rsl-idea-section__range

Mode

The mode of a data set is the most commonly occurring score.

To find the mode, the frequency of each score is determined by counting how many times each score occurred. The mode of the data set is the score with the highest frequency.

Find the mode of the following scores:$$6,1,8,1,6,9,7,6,8$$

2	1
4	1
6	3
5	1
8	1
1	2
9	1
7	1

Here, 6 has the highest frequency. Hence the mode is 6.
Advantage of the mode: It is advantageous over the median and the mean as it can be used for both categorical and numerical data.Disadvantage: The mode may not reflect the centre of the distribution if it has morethan one mode (bimodal, or multi-modal), limiting its ability to describe the centre of a distribution. Also, sometimes there might be no mode if the values are all different.