Measures of Central Tendency: Mean, Median, and Mode

Measures of central tendency help you to find the average, most recurring, or average value in a particular data set. The measure of central tendency indicates where the most values in a set are clustered around. In statistics, there are three main measures of central tendency: mean, mode, and median.

Mean: Also called the average, the mean is the sum of all values divided by the total number of values. For example, the mean of 2 and 3 will be 2+3 /2 = 2.5

Median: The median is the middle term in a data set. The median in the following data set, 2,3,4,5,6, would be 4 since it’s the value in the center.

Mode: The recurring term in a data set. The mode of the following data set, 1,2,2,3, would be 2 since it appears most times.

Now that we have a simple idea of each central tendency measure let’s look at them in a little more detail.

Median

The median of a data set is precisely at the middle of the center term when the set is ordered from the lowest to the highest order.

Let’s look at an example,

Example: Find the median of the following data set

2 , 5 , 7 , 8 , 3 , 6 , 1

Solution:

Step 1: Arrange them in the order from the lowest to the highest like this,

1,2,3,5,6,7,8

Step 2: Now check for the middle term in the correctly ordered data set

1,2,3,5,6,7,8

(The median is 5)

Remember: It’s crucial to arrange the data set from the lowest to the highest term. If you skip this step, you will get a completely different answer. In this case, your answer would have been 8, which is wrong.

The median of the Odd-Numbered Data Set

The basic formula to find out the median for an odd-numbered data set is,

(n + 1) /2 observation

Where n = total number of values in the data set,

Example: Find the median of the following data set,

164 , 222 , 187 , 200 , 101 , 286 , 192

Solution:

Step 1: Arrange them in order from the lowest to the highest.

101,164,187,192,200,222,286

Step 2: Using the formula (n+1)/ 2

(7+1)/2

=8/2 = 4th observation.

Step 3: Check for the 4th term in the ordered data set.

Step 4: 101,164,187,192,200,222,286

Therefore, our median is 192

Median of Even-Numbered Data Set

In an even-numbered data set, the middle term is found by n/2 and (n/2) + 1, where n is the total number of values.

Example: Find the median of the following data set,

164 , 222 , 187 , 200 , 101 , 286

Solution:

Step 1: Arrange them in order from the lowest to the highest.

101,164,187,200,222,286

Step 2: Using the formulas:

 (n/ 2) + 1                                                         n/2

(6/2) + 1                                                          =6/2                                    

=3+1                                                                =3rd observation

= 4th observation

Step 4: Check for the 3rd and the 4th observation.

101,164,187,200,222,286

Step 5: Take the average of these two terms.

(187+200) / 2

=193.5 Answer

Mode

Mode is the most occurring value in a data set. It is possible to have no mode, one mode, or more than one mode in a set,

Example: In a university, you ask the students about their majors. To find the mode easily, you arrange the data in a table. The results are the following,

Mean

Also called the average, the mean is the sum of all values divided by the total number of values. This is the most frequently used measure of central tendency.

The formula of the sample mean is

Sample Mean (x̄)= (⅀x) / n

Where ⅀x is the sum of all the values and n is the total number of values.

Example: Find the sample mean,

100 , 200 , 350 , 400 , 150

Solution:

⅀x= 100+200+350+400+150

⅀x=1200

n=5

Mean (x̄)= (⅀x) / n

Mean (x̄)= 1200/5

Sample Mean (x̄)=240

Population Mean

You may have noticed that we used the term ‘sample mean’ in the above example. This is because sample and population have two different meanings in statistics. Although the method for calculating sample and population mean is the same, the letters in the formula differ to distinguish between them.

For population mean, we use the following formula,

μ = ⅀ X/N

Where,

μ = population mean

⅀x= the sum of all the values in the population data set

n= the total number of values in the population data set.

When Should you use mean, median, and mode

1. Median should be used as a measure of central tendency when:
2. There are fewer extreme values in a data set.
3. There are missing values in the data.
4. The data can be ordered, such as in interval, ratio, and ordinal data.
5. Mean should be used on interval and ratio levels of measurement.
6. The mode can be used when data is nominal, or in some cases, ordinal.

FAQs About Central Tendency