Calculating Mean, Median, and Mode
What are Mean Median and Mode?
Mean, median and mode are all measures of central tendency in statistics. In different ways they each tell us what value in a data set is typical or representative of the data set.
The mean is the same as the average value of a data set and is found using a calculation. Add up all of the numbers and divide by the number of numbers in the data set.
The median is the central number of a data set. Arrange data points from smallest to largest and locate the central number. This is the median. If there are 2 numbers in the middle, the median is the average of those 2 numbers.
The mode is the number in a data set that occurs most frequently. Count how many times each number occurs in the data set. The mode is the number with the highest tally. It's ok if there is more than one mode. And if all numbers occur the same number of times there is no mode.
Example
Mean, Median, and Mode
The mean, median, and mode are measures of central tendency used to summarize a set of data points. Below are their formulas and explanations:
Mean
The mean is the average of a set of numbers. To calculate the mean, sum all the values in the set and divide by the total number of values.
\[ \text{Mean} = \frac{\sum \text{Values}}{n} \]
\[\text{EX:} \quad \frac{5 + 10 + 15 + 20}{4} = \frac{50}{4} = 12.5\]
Median
The median is the middle value in a set of numbers when they are arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle numbers.
\[\text{EX:} \quad \text{For the set } [3, 5, 8, 12, 13], \text{ the median is } 8.\]
\[\text{EX:} \quad \text{For the set } [2, 4, 6, 8], \text{ the median is } \frac{4 + 6}{2} = 5.\]
Mode
The mode is the value that appears most frequently in a data set. A set may have one mode, more than one mode, or no mode at all if all values appear with the same frequency.
\[\text{EX:} \quad \text{For the set } [2, 3, 3, 4, 5], \text{ the mode is } 3.\]
| Problem | Formula | Solution | Explanation |
|---|---|---|---|
| Find the mean of [5, 10, 15, 20] | \( \text{Mean} = \frac{\sum \text{Values}}{n} \) | Solution: \( \frac{5 + 10 + 15 + 20}{4} = \frac{50}{4} = 12.5 \) | Explanation: Sum all the values and divide by the number of values: \( \frac{50}{4} = 12.5 \) |
| Find the median of [3, 5, 8, 12, 13] | For odd number of values, the median is the middle value | Solution: Median = 8 | Explanation: The middle value of the sorted set [3, 5, 8, 12, 13] is 8. |
| Find the median of [2, 4, 6, 8] | For even number of values, the median is the average of the two middle values | Solution: Median = 5 | Explanation: The two middle values of the sorted set [2, 4, 6, 8] are 4 and 6. The median is \( \frac{4 + 6}{2} = 5 \). |
| Find the mode of [2, 3, 3, 4, 5] | Mode = the value that appears most frequently | Solution: Mode = 3 | Explanation: The number 3 appears most frequently in the set, so it is the mode. |
| Find the mode of [5, 5, 7, 7, 9, 9] | Mode = values that appear most frequently | Solution: Mode = 5, 7, 9 | Explanation: Since 5, 7, and 9 each appear twice, they are all modes, making the set multimodal. |
