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.

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‘.