Calculation and Application of Simple Arithmetic Mean, Median, and Mode

Here, we discuss the calculation and application of simple arithmetic mean, median, and mode among the measures of average.
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Calculation and Application of Simple Arithmetic Mean, Median, and Mode

Introduction

After classifying and tabulating the collected data, the next step is data analysis. Various measures of average are useful for data analysis. A measure of average represents a single value that represents the values in a group. 

This value tends to be concentrated in the middle part of the statistical series, hence it is also called the central value. Other values in the group are around the central value or centered around the central value.

Therefore, since the central value or average value lies between the minimum and maximum values of the series and represents all the values in the statistical series, such values are called averages or measures of central tendency. 

Here, we discuss the calculation and application of simple arithmetic mean, median, and mode among the measures of average.

Simple Arithmetic Mean (AM) and Its Calculation

Simple arithmetic mean is also called arithmetic mean or simply mean. Among the measures of average, it is the simplest and most commonly used measure. The simple arithmetic mean is obtained by adding all the values of the variable and dividing by the total number of values or items.

Calculation of Simple Arithmetic Mean

There are mainly three types of series. The calculation process of simple arithmetic mean according to the type of series is presented below:

(a) Individual Series

If the frequency of all values of the variable or item in a series is one, such a series is called an individual series. The following formula is used to calculate the simple arithmetic mean of an individual series:

X̄ = ΣX / N

Where, 

X̄ = Simple arithmetic mean 

ΣX = Sum of the values of the items 

N = Number of items

(b) Discrete Series

If the values of the variable or item in a series are integers or whole numbers and their frequencies are given, such a series is called a discrete series. In such a series, the frequency of all values of the variable or item is not one. The following formula can be used to find the simple arithmetic mean of a discrete series:

X̄ = ΣfX / N

Where, 

X̄ = Simple arithmetic mean 

ΣfX = Sum of the product of frequency and the value of the variable 

N = Σf = Sum of the frequencies

(c) Continuous Series

In a continuous series, the values of the variable or item are in the form of class intervals. The frequencies related to these class intervals are also given. To find the simple arithmetic mean of such a series, the midpoint (mid-value) of each class interval needs to be found. 

There are three methods to calculate the simple arithmetic mean of a continuous series: direct method, short-cut method, and step-deviation method. Although different methods are used, the result obtained will be the same. These methods are discussed separately below.

(i) Direct Method

The following formula is used to calculate the simple arithmetic mean of a continuous series using the direct method:

X̄ = Σfm / N

Where, 

X̄ = Simple arithmetic mean 

m = Midpoint of the class 

f = Frequency 

Σfm = Sum of the product of frequency and midpoint 

N = Σf = Sum of the frequencies

The following steps should be followed to calculate the simple arithmetic mean of a continuous series using the direct method:

  • Find the midpoints (m) of the different classes.
  • Multiply the midpoint (m) of each class by its corresponding frequency (f) and find the sum of the products, which is denoted by Σfm.
  • Divide Σfm by the sum of the frequencies (N = Σf) or use the formula for the direct method to find the simple arithmetic mean (X̄) of the continuous series.

(ii) Short-Cut Method

Although the direct method is simple, if the values of the midpoint and frequency are very large, the values of fm will also be large. In this situation, it becomes a bit difficult to calculate the simple arithmetic mean. 

We can use the short-cut method to overcome this difficulty. The short-cut method is also called the assumed mean method. 

The following formula is used to calculate the simple arithmetic mean of a continuous series using the short-cut method:

X̄ = A + Σfd / N

Where, 

A = Assumed mean 

d = Deviation, i.e., the difference between the midpoint and the assumed mean (m - A) 

Σfd = Sum of the product of frequency and deviation 

N = Σf = Sum of the frequencies 

X̄ = Simple arithmetic mean

The following steps should be followed to calculate the arithmetic mean of a continuous series using the short-cut method:

  • Find the midpoints (m) of the different classes.
  • Assume one of the midpoints as the assumed mean (A). It is preferable to take the middle midpoint as the assumed mean.
  • Calculate the deviation (d) by subtracting the assumed mean (A) from each midpoint (m) (d = m - A).
  • Multiply the frequency (f) of each class by its corresponding deviation (d) to get fd, and find the sum of fd (Σfd).
  • Divide Σfd by the sum of the frequencies (N = Σf) and add it to A, or use the formula for the short-cut method to find the simple arithmetic mean.

(iii) Step-Deviation Method

The following formula is used to calculate the simple arithmetic mean of a continuous series using the step-deviation method:

X̄ = A + (Σfu' / N) × h

Where, 

X̄ = Simple arithmetic mean 

A = Assumed mean 

u' = (m - A) / h 

Σfu' = Sum of the product of u' and f 

h = Common factor of the deviations 

d = Deviation, i.e., the difference between the midpoint and the assumed mean (m - A) 

N = Σf = Sum of the frequencies

The following steps are followed to calculate the simple arithmetic mean of a continuous series using the step-deviation method:

  • Find the midpoint (m) of each class.
  • Assume one of the midpoints as the assumed mean (A). It is preferable to take the middle value among the midpoints. This makes the calculation easier as the values become smaller. If we have six midpoints, it is appropriate to take the third midpoint. Similarly, if there are seven midpoints, it is better to assume the third or fourth midpoint as the assumed mean. However, the result will be the same regardless of which midpoint is taken as the assumed mean.
  • Subtract the assumed mean (A) from each midpoint (m) to find the deviation (d = m - A) and put this in a column.
  • If there is a common factor (h) in each deviation (d), divide each d by h to get u' = d / h and put this in another column.
  • Multiply the frequency (f) of each class by the corresponding u' to get the product fu', and find the sum of these products (Σfu').
  • Divide Σfu' by the sum of the frequencies (N = Σf), multiply the quotient by (h), and add it to (A), i.e., use the formula for calculating the simple arithmetic mean using the step-deviation method.

Median and Its Calculation

When the values of the variable or item in a series are arranged in ascending order or descending order, the middle value of the variable or item is called the median. The exact middle value (median) divides the series into two equal halves. In other words, the number of item values on both sides of the median is equal. The item values on one side of the median are smaller than the median, while the item values on the other side are larger than the median. The median is a positional average.

Calculation of Median

The calculation of the median depends on the nature of the series. The calculation of the median for individual, discrete, and continuous series is mentioned separately below.

(a) Individual Series

The median of an individual series can be calculated as follows:

First, the given data is arranged in ascending order or descending order.

  • Then, the following formula is used:

Median (Md) = Value of the ((N + 1) / 2)^th item

Where, 

  • Md = Median 

  • N = Number of items

When finding the median of an individual series, the following two situations arise:

  • Situation I: If the number of items in the series is odd, the middle item of the series represents the median.
  • Situation II: If the number of items in the series is even, there will be two middle items in the series. The median is found by adding these two items and dividing by 2.

(c) Continuous Series

The following steps are followed to find the median of a continuous series:

First, the cumulative frequency is calculated.

  • The median class is found using the formula:

    Median (Md) = Value of the (N / 2)^th item

    Where, 

    The class in which this item falls is called the median class.

Md = Median 
N = Sum of the frequencies

  • Then, the exact median is found using the following formula:

    Median (Md) = L + ((N / 2) - cf) / f x h

    Where, 

L = Lower limit of the median class 
N = Σf = Sum of the frequencies 
cf = Cumulative frequency of the class preceding the median class 
f = Frequency of the median class h = Class width of the median class

Mode and Its Determination

Generally, the value of the item that is repeated most often or has the highest frequency in a series is called the mode. However, sometimes the value of the item that is repeated most often is not the mode, but some other item value becomes the mode. 

Therefore, the mode is that number where the values of the items are concentrated in the maximum amount. Mode is also called the modal value.

The mode is especially used by factories that produce consumer goods on a large scale, such as garment factories, shoe factories, etc. Such factories maximize their profits by producing items of the most demanded or most fitting size for most people, rather than producing items of the average size.

Determination of Mode

The method of determining the mode according to different series is mentioned below.

(a) Individual Series:

To find the mode of an individual series, it is necessary to find out how many times different items are repeated. The item that is repeated most often is the mode.

If more than one item has the same maximum frequency, then the mode is uncertain, and it is not possible to find the mode directly in such a situation. If two items have the same maximum frequency, such a series is called a bimodal series. 

A series in which more than two items have the same maximum frequency is called a multimodal series. In such a situation, the mode is found indirectly using the following formula:

Mode = 3 x Median - 2 x Mean

(b) Discrete Series:

Generally, to find the mode of a discrete series, the value of the item with the highest frequency is found by inspection, and that becomes the mode. However, the result obtained by inspection may not always be correct. 

Especially if there is not much difference between the maximum frequency and the lower frequency, and if there is an excess of frequencies before and after the lower frequency, the mode determined by inspection may be wrong. 

In such a situation, a grouping table and an analysis table should be used, the use of which will be covered in higher classes. 

In a discrete series also, if there are more than one item with the same maximum frequency upon inspection, the mode can be found indirectly using the following formula:

Mode = 3 x Median - 2 x Mean

(c) Continuous Series:

The following steps can be followed to find the mode from a continuous series:

  • Arrange the items of the series in ascending order.

  • If it is an inclusive series, convert it into an exclusive series.

  • If the class intervals are unequal, make them equal.

  • Find the modal class. Generally, the class with the highest frequency is the modal class. If there is more than one class with the same highest frequency, the following formula can be used:

    Mode = 3 x Median - 2 x Mean

  • Use the following formula to determine the value of the mode:

    Mode (Mo) = L + {(f1 - f0) / (2f1 - f0 - f2)} x h

    Where, 

    This formula can also be written as:

    Mode (Mo) = L + (Δ1 / (Δ1 + Δ2)) x h

    Where, 

Δ1 = f1 - f0 

Δ2 = f1 - f2

L = Lower limit of the modal class 
f0 = Frequency of the class preceding the modal class 
f1 = Frequency of the modal class 
f2 = Frequency of the class succeeding the modal class 
h = Class width of the modal class

Use of Simple Arithmetic Mean, Median, and Mode

Although simple arithmetic mean, median, and mode are all measures of average, their use cannot be the same everywhere. The use of any average depends on the nature of the data and the objective of the study. 

In economics, the use of simple arithmetic mean seems appropriate in some situations for analyzing data, while in other situations, it is appropriate to use the median. 

Similarly, there are some situations where the use of mode becomes necessary instead of simple arithmetic mean or median. 

Here, the use of simple arithmetic mean, median, and mode is briefly mentioned separately.

(a) Uses of Simple Arithmetic Mean: 

Simple arithmetic mean is particularly useful in the following situations:

  • To find the average of data that can be expressed numerically or quantitatively, such as price, wage, income, production, cost.
  • When there is not much skewness in the distribution of data.
  • When there are no open-end class intervals in the distribution.
  • When the values of any item in the distribution are not much larger or smaller than the values of other items.

(b) Uses of Median: 

Median is more useful in the following situations:

  • Median is considered the most appropriate average for data that cannot be expressed numerically, such as beauty, honesty, intelligence, etc.
  • Median can be used in a frequency distribution with open-end class intervals.
  • Median can be used when there is high skewness in the data.

(c) Uses of Mode: 

The use of mode is particularly done in the following situations:

  • Mode is mostly used for taking decisions in business activities such as production and sales. For example, if a shopkeeper wants to buy and keep various items in their warehouse for sale, they choose the items that sell in the highest quantity. Similarly, garment and shoe factories use mode to increase production by determining the ideal size or modal size.
  • Mode is used to study consumer preferences.
  • It is used to find the most common item value in a series.
  • Mode can also be used in a frequency distribution with open-end class intervals.

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