Mean, Median and Mode| Comparison, Relationship and Calculation

A single value used to symbolise a whole set of data is called the Measure of Central Tendency. In comparison to other values, it is a typical value to which the majority of observations are closer. Average and Measure of Location are other names for the Measure of Central Tendency. In statistical analysis, the three principal measurements used in central tendency are Arithmetic Mean, Median, and Mode.

What is Mean?

Arithmetic Mean is one approach to measure central tendency in statistics. This measure of central tendency involves the condensation of a huge amount of data to a single value.

What is Median?

The positional value of a variable known as the median distributes the distribution into two parts that are equal; i.e., values above or equal to the median value are included in the first part, while all values below or equal to the median are included in the second part.

What is Mode?

Mode refers to the variable that occurs most of the time in the given series. In simple words, mode is a variable that repeats itself most frequently in a given series of variables (say, X). Mode is denoted as ‘Z‘.

Comparison between Mean, Median, and Mode

The decision of which approach to employ for a particular collection of data depends on a number of factors that can be grouped into the following major categories:

1. Rigidly Defined: Mean and median are defined rigidly; on the other hand, mode is not always rigidly defined.

2. Based on all Observations: A suitable average should be calculated based on all observations. This attribute is only met by mean and not by median or mode.

3. Possess Sampling Stability: Mean should be preferred when the criteria of least sampling variability is to be attained.

4. Additional Algebraic Treatment: It should be able to get additional mathematical treatment. This attribute can only be satisfied by the mean, hence the majority of statistical theories utilise the mean as a measure of central tendency.

5. Simple to Calculate and Understand: It should be simple to understand and interpret an average. All three averages; i.e., mean, median, and mode satisfy this attribute.

6. Not significantly Impacted by Extreme Values: The appropriate average shouldn’t be significantly impacted by extreme observations. From this perspective, the mode acts as the most appropriate average. The existence of extreme observations has a very small effect on the median but a large impact on the mean.

Relationship between Mean, Median, and Mode

The distribution type—symmetrical or asymmetrical, determines the relationship between the mean, median, and mode.

1. Symmetrical Distribution: For symmetrical curves, Mean (X) = Median (Me) = Mode (Z) because the mean, median, and mode values are all equal in the symmetrical distribution. The symmetrical distribution forms a bell-shaped curve.

The median divides the area of the curve into two equal halves, the Mean is the center of gravity, and the Mode touches the peak of the curve, which indicates maximum frequency.

2. Asymmetrical Distribution: Most distributions in real life are not symmetrical. The mean, median, and mode have different values in an asymmetrical series. Since the curve’s height is not near the middle, the frequency curve is not bell-shaped. Asymmetrical (skewed) distributions can be either positively or negatively skewed.

• When a distribution is positively skewed, the majority of observation values fall to the right of the mode. These measurements will be in the following order of magnitude: Mean>Median>Mode.
• When a distribution is negatively skewed, lower magnitude values are more concentrated to the left of the mode. These measurements will have the following magnitude: Mean<Median<Mode.

The relationship between mean, median, and mode can also be ascertained using the formula. In the case of an asymmetrical distribution, the relationship between the mean, median, and mode is provided by the following formula: 

Mode = 3 Median – 2 Mean

The third value (mean, median, and mode) can be calculated using the provided formula if the other two values are given. Also, if the distribution is moderately asymmetrical, the result will be similar to that obtained by applying the exact formula. It can be better understood with the help of an example.

Example:

The median and mean values of a series that is moderately asymmetrical are 30 and 20, respectively. Determine the mode value.

Solution:

In the above example to calculate the value of mode, the following formula is to be used:

Mode = 3 Median – 2 Mean

Where, Median = 30 and Mean = 20.

Thus, Mode = 3(30) – 2(20)

Mode = 90 – 40

Mode = 50

Which is the Best Average?

There are several ways to measure central tendency, which include the arithmetic mean, median, mode, etc. It cannot be said that one of them is the best average. Different averages are appropriate for various situations. However, the following points must be taken into consideration while choosing the appropriate average:

1. Objective: The average must be chosen in accordance with the study’s objectives. For instance, the arithmetic mean will be most suitable if all values are to be given the same significance. However, the mode will be most helpful if the value that happens most frequently in a series needs to be determined.

2. Number of Variables: When there are few variables in a series, the arithmetic mean is the best indicator of the series’ central tendency.

3. Distribution of Items and Frequency: The arithmetic mean may not be helpful if the value of a large number of items in a series is small but that of one or two things is large. The use of the median should be the ideal choice if most of the values are found in the middle of the series or are associated with qualitative facts.

4. Importance to the Highest and Lowest Items: If the highest and lowest items in a series are not necessary, using median or mode is the best option.

5. Different Types of Series: The use of mode is inappropriate if a large number of the items in a series are identical to one another.

Calculation of Mean, Median, and Mode in Special Cases

In some cases, the calculation of Mean, Median, and Mode is different. Following are some of the treatment of special cases.

Cases	Mean	Median	Mode
Cumulative Series (‘Less than’ or ‘More than’)	Convert the cumulative frequency to a basic frequency distribution before calculating the mean as usual.	Convert the cumulative frequency to a basic frequency distribution before calculating the median as usual.	Calculate mode as usual after converting the cumulative frequency into a basic frequency distribution.
Mid-Values are given	Calculate the mean in the usual way. There is no need to transform mid-values into class intervals.	Calculate the median after converting the mid-values to class intervals.	Calculate the mode after converting the mid-values into class intervals.
Inclusive Class-interval	Compute the mean as usual. The series should not be changed into an exclusive class-interval series.	To determine the median, the class interval is transformed into an exclusive class-interval series.	After converting class intervals into exclusive class interval series, the mode is determined.
Open-End Series	In order to calculate the mean, missing class limits will be assumed, and this depends on the distribution of class intervals among other classes.	The median is calculated using the normal procedure without completing the class intervals.	Mode is computed as usual, without completing the class intervals.
Unequal Class-intervals	In this case, after calculating the mid-values for each interval, the mean may be calculated as usual. There is no need to make equal class-intervals.	The median can be determined in the usual manner regardless of unequal class intervals.	Class intervals and frequencies need to be adjusted in order to calculate mode.