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Unit 4: Data, Statistics, and Probability

Topic 2

Mean, Median, Mode, and Range

For a data set x1,x2,…,xnx_1,x_2,\dots,x_n:
  • Mean (average): xˉ=x1+⋯+xnn\displaystyle \bar{x}=\frac{x_1+\cdots+x_n}{n}.
  • Median: middle value after sorting. If nn is even, average of the two middle values.
  • Mode: most frequent value(s).
  • Range: max minus min.
Outliers pull the mean more than the median. Median is more resistant to extreme values.

Core Skills

  • Sort the data before finding the median and range.
  • Compute mean quickly using totals or by using balance reasoning.
  • Identify modes from a list or simple frequency table.
  • Decide which measure best represents the center given outliers or skew.

Example 1: All Four Measures

Data: 7,9,4,6,97, 9, 4, 6, 9. Sorted: 4,6,7,9,94,6,7,9,9.
xˉ=355=7,median=7,mode=9,range=9−4=5.\bar{x}=\frac{35}{5}=7,\quad \text{median}=7,\quad \text{mode}=9,\quad \text{range}=9-4=5.

Example 2: Even Count Median

Data: 12,15,18,2012, 15, 18, 20. Sorted already. Median =15+182=16.5=\frac{15+18}{2}=16.5. Mean =654=16.25=\frac{65}{4}=16.25. Range =20−12=8=20-12=8. No mode.

Example 3: Effect of an Outlier

Test scores: 70,74,76,78,80,82,10070, 74, 76, 78, 80, 82, 100. Mean =5607=80=\frac{560}{7}=80. Median is the 4th value =78=78. The 100 raises the mean above the median. Median better represents the typical score.

Example 4: From a Frequency Table

Value2345Frequency1232\begin{array}{c|cccc} \text{Value} & 2 & 3 & 4 & 5\\ \hline \text{Frequency} & 1 & 2 & 3 & 2 \end{array}
Total count n=8n=8. Sum =2(1)+3(2)+4(3)+5(2)=2+6+12+10=30=2(1)+3(2)+4(3)+5(2)=2+6+12+10=30. Mean =30/8=3.75=30/8=3.75. Mode =4=4 (highest frequency). Sorted positions 4th and 5th are both 4, so median =4=4. Range =5−2=3=5-2=3.

Example 5: Shifts and Scales

If each data value increases by kk, mean and median increase by kk, range unchanged. If each data value is multiplied by c>0c>0, mean, median, and range are multiplied by cc.

Key Takeaways

  • Use the median when data are skewed or have outliers.
  • Mean equals total divided by count. Keep track of nn.
  • Mode describes most common value. There can be more than one or none.
  • Range measures spread but is sensitive to extremes.