Measures of Central Tendency for Ungrouped Data

Measures of central tendency, also known as measures of center, are statistical measures that represent the “typical” or “central” value of a dataset. They provide a single value that summarizes the entire dataset. For ungrouped data (raw data), we calculate these measures directly from the individual data points.

The three primary measures of central tendency for ungrouped data are the mean, median, and mode.

Formulae

Mean (Arithmetic Mean): The average of all the data points.

Formula: $Mean = \frac{\sum_{i=1}^{n} x_i}{n}$ where $x_i$ represents each data point and $n$ is the total number of data points.

Median: The middle value when the data is arranged in ascending order.

If n is odd: $Median = x_{\frac{n+1}{2}}$
If n is even: $Median = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2}$ (average of the two middle values)

Mode: The value that appears most frequently in the dataset. A dataset can have no mode, one mode (unimodal), two modes (bimodal), or more than two modes (multimodal).

Examples

Example-1: Consider the following dataset representing the scores of 5 students on a math quiz: 70, 80, 80, 90, 100.

Mean: $\frac{70 + 80 + 80 + 90 + 100}{5} = \frac{420}{5} = 84$
Median: The data is already sorted. Since there are 5 data points (odd number), the median is the middle value: 80.
Mode: The value 80 appears twice, more than any other value. So, the mode is 80.

Example-2: Consider the following dataset: 2, 4, 5, 5, 6, 8.

Mean: $\frac{2 + 4 + 5 + 5 + 6 + 8}{6} = \frac{30}{6} = 5$
Median: The data is already sorted. Since there are 6 data points (even number), the median is the average of the two middle values: $\frac{5+5}{2} = 5$.
Mode: The value 5 appears twice, more than any other value. So, the mode is 5.

Common mistakes by students

Miscalculating the Mean: Forgetting to divide by the total number of data points.
Incorrectly Sorting for the Median: Failing to arrange the data in ascending order before identifying the median.
Misinterpreting the Mode: Confusing the value that appears most often with the *number* of times a value appears. Students might incorrectly state the mode as the count (frequency) instead of the data value itself.
Forgetting to Average for Even n (Median): When finding the median with an even number of data points, students sometimes forget to take the average of the two middle numbers.

Real Life Application

Measures of central tendency are used extensively in various real-life scenarios:

Analyzing Test Scores: Teachers use the mean, median, and mode to understand the overall performance of a class on an exam.
Income Analysis: Economists use the median income to get a more accurate representation of typical earnings, as the mean can be skewed by extremely high earners.
Market Research: Businesses use these measures to understand consumer spending patterns. For example, the mode could indicate the most popular product price point.
Sports Statistics: Athletes and coaches use these measures to analyze performance data (e.g., mean points per game, median time to run a mile).

Fun Fact

The mean is sensitive to outliers (extreme values), while the median is not. This is why the median is often preferred when analyzing data that might contain extreme values.