CBSE · Class 10 · Mathematics · StatisticsExplain the three main measures of central tendency (Mean, Median, and Mode). Discuss their definitions, formulas, and conditions under which one measure is preferred over the others in statistical data analysis.

1. Introduction to Measures of Central Tendency\nIn statistics, a measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. The three most commonly used measures are the Mean, Median, and Mode.

2. Mean (Arithmetic Mean)

• Definition: The mean is the sum of all observations divided by the total number of observations. It represents the mathematical average.
• Formula: For grouped data, the direct method formula is $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$, and the assumed mean method is $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$.
• Characteristics: It takes into account all values in the data set. However, it is heavily influenced by extreme values (outliers).

3. Median

• Definition: The median is the middle value of a sorted data set. For grouped data, it divides the distribution into two equal parts.
• Formula: $\text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h$, where $l$ is the lower limit of the median class, $N$ is total frequency, $cf$ is cumulative frequency of preceding class, $f$ is frequency of median class, and $h$ is class size.
• Characteristics: It is not affected by extreme values, making it a better measure when extreme outliers are present.

4. Mode

• Definition: The mode is the value that appears most frequently in a data set. In grouped data, it is the value associated with the maximum frequency.
• Formula: $\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$, where $l$ is lower limit of modal class, $f_1$ is frequency of modal class, $f_0$ and $f_2$ are frequencies of preceding and succeeding classes respectively, and $h$ is class size.
• Characteristics: Useful for finding the most popular item or standard size.

5. Preference Conditions

• Use Mean when: The data is symmetrical without extreme outliers, and every data point needs to contribute equally to the final summary.
• Use Median when: There are extreme values or outliers in the dataset, or when dealing with open-ended distribution intervals.
• Use Mode when: You need to find the most typical, recurring, or popular category/item (e.g., standard shoe size manufactured).