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What are Measures of Central Tendency?

Measures of central tendency can be seen as one of the most useful and basic statistical function. You all might have a question as to what is the main purpose of using measures of “central tendency”. The most basic as well as easy answer to this is that it concludes a population or sample by a single value. Measures of central tendency are basically numbers that tells you what is average within a distribution of data. It is a measure that describes a whole set of data with a single value that indicates the middle value of the given distribution data.Therefore, measures of central tendency are also called as measures of central location.

One of the important objectives of measuring central tendency is to find a single figure which will be able to represent a whole series of different variables.

What are the different measures of central tendency?

There are three main measures of central tendency which are calculated differently as well as measures something different from the others under different circumstances.These different methods of measuring “central tendency” gives us different types of averages as well as different indication of the typical value in the data distribution.Three most important measures of central tendency for numerical data that are commonly used are as follows:
1. Mean
2. Median
3. Mode

Below, we will look at the mean, mode and median, and learn how they can be calculated and under what conditions they are most appropriate to be used.

1.What is mean (arithmetic)?

You all must be familiar with the mean which is often known as the arithmetic average. The arithmetic mean is considered as the most common measure of central tendency. It is mostly used with continuous data although it could be used with both discrete and continuous numeric data.

A mean is very easy to calculate.The mean is equal to the sum of the observations in a data set divided by the number of total observations in a data set. Suppose, we have n values in a data set and the whole data set has values such as  x1, x2, …, xn then the sample mean , usually indicated by “x̅” (pronounced X-bar)  is:

This formula is often written in a different manner using the Greek capital letter, “Σ”,( pronounced sigma), which means “sum of”:

where ΣX is the sum of all the values in the sample and n is the number of values in the sample.

Did you notice that the above formula refers to the sample mean? So, the important question is why have we called it a sample mean? This is because, in statistics, samples and populations have very different meanings but in the case of the mean, they are calculated in the same way. To indicate that we are not calculating the sample mean but the population mean, we use the Greek symbol “μ”(pronounced ‘mu’)

The formula for μ is shown below:

where ΣX is the sum of all the observations in the population and n is the number of observations in the population.

For example, if four families have 0, 2, 3, and 5 children respectively, then the mean number of children is (0 + 2 + 3 + 5)/4 = 10/4 = 2.5. This means that the five households have an average of 2.5 children.

Advantage of the mean: 

One of the important advantages of arithmetic means is that it can be used for both continuous and discrete numeric data.

Limitations of the mean: 

It is not possible to calculate mean for categorical data as it is difficult to sum those values.Also, this direct method of mean cannot be used where values are extremely big as adding together all the values will be a lengthy process. The mean is easily  influenced by skewed and outlier distributions because it takes every value present in the distribution. And the values present in these distributions are unusual as compared to the rest of data set because they are either too small or large.

2.What is the median? 

The median is the middle value/score we get in a distribution after the values are arranged in ascending or descending order. It is the midpoint of a distribution The median divides the sets of data in half (there are 50% of observations on either side of the median value).

In order to calculate the median, suppose we have retirement age distribution as below:

    60     55     54      56      58     52    55     53     62

We first need to rearrange that data into order of magnitude that is the smallest digit will come first:

    52      53     54     55     55      56      58     60     62

Looking at the retirement age distribution which has 9 observations You will see that our median value is the middle mark – in this case, 55 (highlighted in bold). It is the middle value because there are 4 values before it and 4 values after it. Through this observation, you can conclude that whenever there is a distribution with an odd number of observations , the median will always be the middle value.

This method is easy when you have odd number of observations but what if you have even number of observations.In that case,the median value will be the mean of the two middle values.Let us look at age of 10 employees:

     25      27     45      39      49      40    55     43     35      28

We again rearrange that data into order of magnitude (smallest first):

     25      27     28      35      39      40     43     45     49      55

As you can see, in the following distribution, the two middle values are 39 and 40 (highlighted). After finding the average of this  5th and 6th values in our data set, you will get a median of 39.5

Advantages of the median: 

One of the important advantages of median is that it is less affected by skewed and outliers data unlike mean.And when you don’t have the symmetrical distribution then median is the preferred measure of central tendency.

Limitation of the median:

You cannot identify the median for categorical data as it cannot be logically ordered.

3.What is the mode? 

The value that occurs most commonly in a distribution is known as the mode. It is the most frequent value in our distribution.Let us consider this set of data showing the number of students in 11 different classes of a school:

   54    55   54    54     56    57     57     58    58     60     60

This table shows a simple frequency distribution of the number of students:

     Age   Frequency
       54        3
       55        1
       56        1
       57        2
       58        2
       60        2

The most commonly occurring value is 54, therefore the mode of this distribution is 54 students.

If you look at this histogram below then you will see that the mode represents the highest bar in a bar chart or histogram:

Limitation of the mode: 

Mode is not preferred when you have continuous data because you will not be able to  find any value that is more frequent than the other. However, another problem arises when two or more values share the highest frequency in a distribution.

Find out more how-to-study articles like Bank Reconciliation Statement and study tips for your board exams here!

 

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