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Mean, Median, Mode and Range

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Why do we study mean, median, mode and range

In this lesson we will learn about mean, median, mode and range. In math, we mostly deal with datasets. Datasets are collection or group of numbers. These group of numbers can be a result of a scientific experiment or the marks of students in a subject. These datasets can be smaller or bigger. When these numbers are smaller, for example, the ages of all family members, we can represent them individually.

But, when the dataset contains hundreds or thousands of numbers, it is quite difficult to talk about each number separately. Therefore, for this purpose, mean, median and mode are used. They all represent or give information about a dataset. More specifically, they measure the center of a group of numbers. Therefore, they are known as measures of central tendency.

Let’s now talk about each one of them separately.

What is mean

Yo might not have heard about mean but you must have heard about and calculated average. Mean and average are the same. Mean is just another word for average. Mean or Arithmetic Mean is defined as sum of the all the observations (or simply numbers) in a dataset divided by the total number of observations. i.e.

mean = sum of observations/total number of observations

or if xi is ith data point in the dataset and n is the total number of data points. Then, Formula of the mean can be written as:

mean =  Σxi / n

where i is from 1 to n.

Meaning of Median

Median is defined as the middle point of the dataset.

Median can be found as follows

  1. Arrange the given numbers in the ascending order i.e. in the increasing order.

  2. If the total number of elements in the dataset is odd, then finding the median is easy. The median, in this case, is middle number in the dataset. For example, if the dataset is {3,4,5,7,4}. Median is 5.

  3. If the total number of elements in the dataset is even. First, find the middle two elements and then take its arithmetic mean or average, as learnt above. For example, if the dataset is {3,4,3,4,7,8}. The middle two elements are 3 and 4. Taking its average, the median will be 3.5.

Finding median manually as explained in step 2 and 3 can be bit tedious when we deal with large number of elements. It is inefficient and also prone to errors. However, we can also perform step 2 and 3 using formulas as given below.

If n is the total number of elements. Then,

  • Formula of median for odd number of elements

median = [(n+1) / 2]th term in the dataset

  • Formula of median for even number of elements

median  = [(n/2)th term + ((n / 2)+1)th term] / 2

What is Mode

Mode is the most frequent or common number(s) in the given dataset. For example, in the dataset {2, 3, 4, 5, 4, 6, 4}, 4 is the most common. Hence, 4 is the mode.

Remember, a dataset can have

  • 1 mode

  • More than one mode

  • No mode at all.

A set has no mode when all the numbers appear only once. For example, consider{2,4,5,6}. Since all the numbers appear only one time this dataset has no mode.

What is range

Range is defined as difference between the lowest and highest values. For example, in {6,4,3,2,5} the range is 6 – 2 = 4.

So, the formula of range can be written as:

Range  = Highest value – Lowest value

However, range can provide misleading values. For example, in {105, 6, 5, 9, 8} the range is 105 – 5 = 100 but the values in the dataset are near 10.

Let’s illustrate the working of formulas by the following examples.

Questions on mean, median, mode and range

Questions on mean

Example 1

Find the mean of 2, 1, 3.

Solution

Mean is found by adding all the numbers and then dividing by the total number of elements in the group. i.e.

mean = (2 + 1 + 3) / 3

mean = 6 / 3 = 2

Example 2

Find the mean of the following dataset.

3, 4, 66, 4, 4, 3, 2, 7, 43, 56

Solution

mean = (3 + 4 + 66 + 4 + 4 + 3 + 2 + 7 + 43 + 56)

mean = 192 / 10

mean = 19.2

Example 3: Student A gets 19, 15, 18, 15 marks in the first four tests of Math where the total marks in each test are 20. How much should A score in the 5th test to get the average of 17?

Solution

Let the marks of the 5th test be x.

We know

mean = sum of all observations / total number of observations

17 = (19+15+18+15+x) / 5

17×5 = (19+15+18+15+x)

85 = 67 + x

x = 85 – 67

x = 18

A should score 18 in his last test to get the average of 17.

Let’s now move on to median now.

Questions on median:

Example 1

Find the median the of the following dataset.

4, 1, 3, 5, 6, 8, 9

Solution

Step 1: Arrange the elements in the ascending order. We get,

1, 3, 4, 5, 6, 8, 9

Step 2: Since the number of elements are odd. The median will be the middle element i.e. 5.

Alternate Approach

Step 2: We can also find median using the following formula

median = [(n+1) / 2]th term (for odd number of elements)

Here, n  = 7

median = [(7+1)/2]th term

median = [(8)/2]th term

median  = 4th term in the dataset

4th term in the dataset is 5. Hence, the median is 5.

Example 2

Find the median of the following group of numbers.

9, 1, 5, 3, 6, 7, 3, 6

Solution

Step 1: Arrange the elements in the ascending order. i.e.

1, 3, 3, 5, 6, 6, 7, 9

Step 2: Since the number of elements are even. Using formula,

median  = [(n/2)th term + ((n / 2)+1)th term]/2

Here n = 8

median  = [(8/2)th term + ((8 / 2)+1)th term]/2

median  = [4th term + 5th term]/2

median  = (5+6)/2

median = 5.5

Hence, the median is 5.5.

Example 3

Find the median of the following group of numbers.

6, 4, 3, 8, 2, 9

Solution

Step 1: Arrange the elements in the ascending order. i.e.

2, 3, 4, 6, 8, 9

Step 2: Since the number of elements are even. Using formula,

median  = [(n/2)th term + ((n / 2)+1)th term]/2

Here, n = 6

median  = [(6/2)th term + ((6 / 2)+1)th term]/2

median  = [3th term + 4th term]/2

median  = (4+6)/2

median = 5

Hence, the median is 5.

Hence median can or cannot be a member of the dataset.

Questions on mode:

Example 1

Find the mode of the following dataset.

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

Solution

In the given dataset:

3 appears 1 time, 4 appears 2 times, similarly 5, 7, 6 and 2 are coming 3, 2, 2 and 1 times respectively.

This dataset has only one mode i.e. 5 which appears 3 times and is the most popular.

There are a lot of chances of making mistakes using this method. Another better trick is to first arrange the numbers and find the mode. i.e.

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

Here we can easily observe that 5 is the most frequent.

Example 2

Find the mode of 11, 23, 33, 15, 11, 23, 34, 20.

Solution

Arranging the numbers in ascending order i.e.

11, 11, 15, 20, 23, 23,  33, 34

Here 11 appears 2 times and 23 also occurs 2 times. Hence, both 11 and 23 are modes.

However, according to some textbooks, the above example will have no mode, because it does not contain a single popular element. Therefore, it depends upon the textbook or teacher to allow multiple modes or not.

Example 3

Find the mode of following dataset.

44, 33, 23, 47, 55, 45

Solution

Arranging the numbers. i.e.

23, 33, 44, 45, 47, 55

Each number appears only once. Hence, no mode exists.

Mixed Problems

Problem 1: The marks obtained by the student A in the 5 five tests of English are 89, 90, 70, 90, 80.The total marks are 100. Find the mean, median, mode and range.

Solution

Mean

Mean can be found as follows:

mean = (89+90+70+90+80) / 5

mean = 419/5

mean = 83.8

Median

Step 1: Arrange the elements in the ascending order. i.e.

70, 80, 89, 90, 90

Step 2: Since the total number of elements are odd. Using formula

median = [(n+1) / 2]th term (for odd number of elements)

Here, n  = 5

median = [(5+1)/2]th term

median = [(6)/2]th term

median  = 3th term in the dataset

3th term in the dataset is 89. Hence, the median is 89.

Mode

Arranging the numbers in the ascending order. We get

70, 80, 89, 90, 90

90 is the most popular i.e. it appears 2 times. Hence, the mode is 90.

Range

The highest value in the data set is 90 and the lowest value is 70.

Hence,

range  = highest – lowest

range = 90 – 70

range =  20

Empirical Relation between Mean, Median, and Mode

Though it is slightly advance topic but interesting too. We can find mean, median and mode separately whenever we are given a set of observations. For example, marks of students in a Math test are 19, 10, 15, 20, 16, 15. From this data, we can find mean, median or mode. Now, let’s consider an example where we are given mode and median and we want to find the mean. How will we find it? Is it possible? Yes and we can find it very easily.

We can understand the relationship between mean, median and mode through frequency distribution graphs. A frequency distribution graph is used to organize data so that we can analyze and understand the data easily.

Symmetrical curve

If the frequency distribution graph has a symmetrical frequency curve then, in this case, mean, median and mode are equal.

symmetric distribution curve showing mean median and mode
The relation between mean, mode and median in symmetric distribution is

Mean = Median = Mode

However, if a frequency distribution graph has an asymmetrical frequency curve i.e. it is skewed distribution then two cases exist.

Positively skewed curve

If a frequency distribution graph is positively skewed then the mean is going to be pulled to the right of the median and mode to the left of the median i.e. mean > median > mode.

positive skewed gaussian curve


Negatively skewed curve

If a frequency distribution graph is negatively skewed then the mean is going to be pulled to the left of the median and mode to the right of the median i.e. mode > median > mean.

negative skewed gaussian curve

The empirical relation between mean, median, and mode in moderately skewed distribution is
                        Mean – Mode  = 3(Mean – Median)

Example 1

Consider a moderately skewed distribution in which the mean is 60.4 and median is 44.5. Find an approximate value of mode. Also, tell if it is positively or negatively skewed distribution.

Solution

Mean = 60.4
Median = 44.5
let Mode = x

The mode for the moderately skewed distribution can be found as follows

Mean – Mode  = 3(Mean – Median)

60.4 – x = 3(60.4 – 44.5)
60.4 – x = 3(15.9)
60.4 – x = 47.7
x = 60.4 – 47.7
x = 12.7

Approximate value of mode = 12.7.
Since mean > median > mode then it is a positively skewed distribution.

Example 2

Consider a histogram for a normal distribution whose mean is 45.6. Find the approximate value of median and mode.

Solution
A histogram is a frequency distribution for the grouped data and a normal distribution is a symmetric distribution where the curve is bell-shaped.
We know that in symmetric distribution the mean, median and mode are equal.
So, if mean = 45.6. Then,
mode = 45.6 and median = 45.6.

Example 3

In a moderately skewed distribution, if median is x, mode is 2 times x and mean is 3 less than median. What is the median?

Solution

Median = x
mode = 2x
mean = x – 3
For moderately skewed distribution, we know that
Mean – Mode  = 3(Mean – Median)
(x – 3) – (2x) = 3[(x – 3) – x]
x – 3 – 2x = 3(x – 3 – x)
-x – 3 = 3(-3)
-x – 3 = -9
x = 9 – 3

x = 6

Median = 6