The mean, median, and mode are measures of central tendency in statistics. They are used to represent various values of a given data set. Here are the formulas for each:
- Mean: The mean is the sum of all the values in the data set divided by the number of values in the data set. For ungrouped data, the mean formula is defined as the sum of the observations divided by the total number of observations. The formula for the mean is:
$$\overline{x} = \frac{\sum_{i=1}^{n}x_i}{n}$$
- Median: The median is the middle number in a data set when the numbers are listed in either ascending or descending order. For odd numbers of observations, the median is the middle value of the given ungrouped data if the data is arranged in ascending order. For even numbers of observations, the median is the average of the two middle values. The formula for the median is:
$$\widetilde{x} = \begin{cases} x_{\frac{n+1}{2}} & \text{if } n \text{ is odd} \ \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2} & \text{if } n \text{ is even} \end{cases}$$
- Mode: The mode is the value that appears most often in a data set. The formula for the mode is:
$$\text{Mode} = \text{Value with the highest frequency}$$
It is important to note that the mean, median, and mode are not always the same value. They can be used together to give a more complete picture of a data set.
