NCERT · Class 10 · Mathematics · StatisticsExplain the empirical relationship between the three measures of central tendency: Mean, Median, and Mode. Also write the mathematical formula representing this relationship.

In statistics, when dealing with moderately skewed distributions, there exists a very important and useful empirical relationship among the three primary measures of central tendency, which are the mean, the median, and the mode. This relationship allows us to approximate any one of these three measures if the other two are already known. Karl Pearson studied this relationship extensively and stated that for a moderately asymmetrical frequency distribution, the difference between the mode and the median is approximately twice the difference between the median and the mean. Mathematically, this is expressed as:

$$\text{Mode} = 3(\text{Median}) - 2(\text{Mean})$$ \nAlternatively, this equation can also be rearranged to find the median or the mean if the other parameters are given. For instance, $3(\text{Median}) = \text{Mode} + 2(\text{Mean})$, or $\text{Median} = \frac{\text{Mode} + 2(\text{Mean})}{3}$. This empirical relationship is widely applied in various statistical problems in class 10 mathematics when data is somewhat irregular and direct calculation of all three measures is time-consuming or when missing values need to be determined quickly.