Mean and median
Use a mean for a total-based summary and a median when resistance to extreme magnitudes matters. Compare the distribution and the purpose, rather than treating either statistic as always better
- Order the data for a median Arrange the values from least to greatest before using their positions \(\displaystyle x_1\le x_2\le\cdots\le x_n\)
- Use the middle position For an odd count, use the middle position in the sorted list, then report the value at that position \(\displaystyle n=7\;\Rightarrow\;\frac{n+1}{2}=4\)
- Average two middle values For an even number of values, average the values at positions n divided by 2 and n divided by 2 plus 1. \(\displaystyle n=6\;\Rightarrow\;\operatorname{median}=\frac{x_3+x_4}{2}\)
- Divide by the count The mean is the sum of all data values divided by the total number of data values \(\displaystyle \bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}\)
- Frequency tables Multiply each value by its frequency for the total; add frequencies for the count. Locate median positions using cumulative counts \(\displaystyle \bar{x}=\frac{\sum xf}{\sum f}\)
Worked example
Six quiz marks are 13, 6, 20, 9, 14 and 8. Find the mean correct to 1 decimal place and the median
- \(\displaystyle 6,\ 8,\ 9,\ 13,\ 14,\ 20\)Order the six marks before locating the median
- \(\displaystyle \begin{gathered}6+8+9+13+14+20\\=70\end{gathered}\)Add all six marks to obtain the total
- \(\displaystyle \bar{x}=\frac{70}{6}=\frac{35}{3}\approx11.7\)Divide the total by 6 and round the mean to 1 decimal place
- \(\displaystyle \operatorname{median}=\frac{9+13}{2}=11\)The two middle ordered marks are 9 and 13, so their average is 11.
Common mistakes
Difficulty levels
Questions about this topic
Why must the data be ordered before finding a median?+
The median is defined by the middle position in an ordered list. The middle item in an unsorted list has no special statistical meaning
When is the median preferred to the mean?+
The median is usually preferred for skewed data or data with outliers because it is less affected by extreme values
What is the difference between the mean and median?+
The mean is the total divided by the number of values. The median is the middle value, or the average of the two middle values, after ordering
How do zeros affect a mean?+
A zero contributes nothing to the total but still counts as an observation in the divisor
How is a median found from a frequency table?+
Add frequencies to find the count, then use cumulative frequencies to locate the central position or pair of positions
Is the median always the best summary?+
No. The purpose matters. A mean links directly to the total; a median is more resistant to extreme magnitudes. Neither describes shape and spread by itself
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics