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Measures of Central Tendency

Measures of Central Tendency

Let's look at some definitions that are necessary for understanding different measures of central tendency.

What is central tendency ?

A measure of central tendency attempts to describe a dataset through a singular value. This singular value is meant to represent the center point or typical value in a dataset. There are three measures of central tendency we need to know about, mean (also referred to as average), median and mode.

Mean

The mean is the measure of central tendency that you should be most familiar with. The process to find the mean is to sum all the values of the data set, and then divide by the number of data points.

Find the mean value of rainfall for the days listed below

Day

1

2

3

5

9

10

Rainfall (mm)

10

0

5

2

29

1

Solution

The mean is given by the sum of all the values divided by the number of values. The sum of values is 10 + 12 + 0 + 5 + 17 + 2 + 29 + 1 + 4 + 14 = 94, and there are 10 values, so the mean rainfall for the ten days is given as 9.4mm.

Median

When we have a set of data that is able to be ordered in some way, we can find the median. The process to find the median is as follows:

Step 1: Order the data, from smallest to largest.

Step 2: If the number of data points is odd, the middle number is the median, meaning we take the value

Step 3: If the number of data points is even, then we take the mean value of the middle two values. This means we take mean of the and value.

Find the median of the following data.

12, 3, 4, 7, 19, 13, 4, 8, 81

Solution

The first thing we need to do is order the data from smallest to largest, and this results in

3, 4, 4, 7, 8, 12, 13, 19, 81

As this has an odd number of data points, the median is the middle number of the ordered dataset, giving a median of 8.

Given below are the heights of 30 children in a class (height given in cm). Find the median height.

168, 172, 151, 145, 181, 162, 174, 159, 149, 180, 164, 171, 150, 143, 189, 167, 176, 156, 144, 186, 166, 177, 153, 140, 184, 163, 178, 158, 149, 187.

Solution

First of all, we must order the data from smallest to largest. We get:

140, 143, 144, 145, 149, 149, 150, 151, 153, 156, 158, 159, 162, 163, 164, 166, 167, 168, 171, 172, 174, 176, 177, 178, 180, 181, 184, 186, 187, 189.

As thirty is even, to find the median we find the mean of the fifteenth and sixteenth values. The fifteenth value is 164, and the sixteenth value is 166. The mean of these values is , meaning the median value is 165.

Mode

The mode of a set of data is the most common value in the dataset. If there are two or more values which are most common, both of these values are the mode.

Find the mode of the following data set.

1, 2, 3, 4, 4, 5, 6, 6, 6, 6, 7

Solution

The mode here would be 6, as this appears four times, which makes it the most common value.

Find the mode of the following numbers.

1, 2, 2, 3, 3, 3, 5, 7, 7, 7, 9, 11, 134

Solution

Both 3 and 7 appear three times, making them both the most common value, meaning the mode is 3 and 7.

Choosing suitable measures of central tendency

Each measure of central tendency has its own advantages and disadvantages.

For the mean, the advantages are that it uses all of the data, and is, therefore, representative of all the data. However, there are disadvantages to using the mean. It is disproportionately influenced by extreme values, which can throw the mean. The mean also cannot be used if our data isn't numerical, and takes the most computation out of all our measures of central tendency.

For the mode, the advantages are that we can find the mode of a set of data, be it numeric or otherwise. There is also limited computation, as we only need to tally the data, meaning if our data comes pre-tallied then this aids the mode. However, a downside is that the mode doesn't necessarily exist. In addition, we can have multiple modes, which doesn't help us describe a lot about the data set. As well as this, the mode doesn't take into account the full data set.

Our final measure of central tendency is the median. The advantages are that the median isn't affected by any outliers or extreme values, and we have very little calculation to do. On the flip side, it does require us to order the set of data, which for large sets of data, is lengthy and time-consuming. It also doesn't take into account the full set of data, which means this could bring in weak results.

Measures of Central Tendency - Key takeaways

  • To find the mean we add up all the values in the data set and divide by the number of data points.

  • The mode is the most common value in a data set.

  • The median is the central value of the data set.

Frequently Asked Questions about Measures of Central Tendency

The three main measures of central tendency are mean, median and mode.

Central tendency is the typical value for a set of data. We can choose different measures for this measure, which are mainly mean, median and mode.

The three main measures of central tendency are mean, median and mode.

Central tendency is the typical value for a set of data. We can choose different measures for this measure, which are mainly mean, median and mode.

A measure of central tendency describes a data set through a singular value, whereas variability describes the dispersion of your dataset.

Final Measures of Central Tendency Quiz

Question

Find the mean of the following set of data: 1,2,2,3,3,5,5,5,6,9,113

Show answer

Answer

14

Show question

Question

Find the mode of the following set of data: 1,2,2,3,3,5,5,5,6,9,113

Show answer

Answer

5

Show question

Question

Find the median of the following set of data: 1,2,2,3,3,5,5,5,6,9,113

Show answer

Answer

5

Show question

Question

When can mean not be used?

Show answer

Answer

When the data is non-numerical

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Question

Find the mean of the following data, which is shoe size in a class.

Shoe Size         Frequency

2                         4

3                         4

4                         5

5                         7

6                         8

7                         10

8                         2


Show answer

Answer

5.225

Show question

Question

Find the mode of the following data, which is shoe size in a class.

Shoe Size         Frequency

2                         4

3                         4

4                         5

5                         7

6                         8

7                         10

8                         2

Show answer

Answer

7

Show question

Question

Find the median of the following data, which is shoe size in a class.

Shoe Size         Frequency

2                         4

3                         4

4                         5

5                         7

6                         8

7                         10

8                         2

Show answer

Answer

5.5

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Question

Find the mean of the following lengths: 12cm, 14cm, 14cm, 56cm, 76cm, 1.24m, 167cm

Show answer

Answer

66.1cm

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Question

Find the mode of the following lengths: 12cm, 14cm, 14cm, 56cm, 76cm, 1.24m, 167cm

Show answer

Answer

14cm

Show question

Question

Find the median of the following lengths: 12cm, 14cm, 14cm, 56cm, 76cm, 1.24m, 167cm

Show answer

Answer

56cm

Show question

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