U19 Measures of Central Tendency (I) - Essential Formulas

This section covers the fundamental measures of central tendency: mean, median, and mode. Understanding these concepts is crucial for summarizing and interpreting data sets.

1 Mean (Arithmetic Mean)

Definition and Formula

The mean, often called the average, is the sum of all data values divided by the number of values. For a data set with $n$ values $x_1, x_2, \dots, x_n$, the mean $\bar{x}$ is given by:

$$ \bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n} $$

If the data is grouped into a frequency table with values $x_i$ and corresponding frequencies $f_i$, the mean is calculated as $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.

2 Median

Finding the Middle Value

The median is the middle value when the data is arranged in ascending or descending order. It is not affected by extreme values (outliers).

For an odd number of data points ($n$ is odd): The median is the value at position $\frac{n+1}{2}$.

For an even number of data points ($n$ is even): The median is the average of the values at positions $\frac{n}{2}$ and $\frac{n}{2}+1$.

$$ \text{Median} = \begin{cases} x_{\left(\frac{n+1}{2}\right)} & \text{if $n$ is odd} \\ \frac{1}{2} \left( x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)} \right) & \text{if $n$ is even} \end{cases} $$

3 Mode

The Most Frequent Value

The mode is the data value that occurs with the highest frequency. A data set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode if all values occur with the same frequency.

For grouped data presented in a frequency table, the modal class is the class interval with the highest frequency.

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