Summarizing Data

Why this matters

A list of 30 numbers tells you almost nothing at a glance. Summarizing turns that list into a few values you can actually reason about: a typical value (centre) and how scattered the data is (spread). Together they let you compare two datasets fairly.

The idea

Organizing first. A frequency distribution table groups values into equal classes and counts the frequency in each. Drawn as bars, that table is a histogram.

Centre: measures of central tendency. One number standing in for the whole set:

• Mean: add everything, divide by the count (spreadsheet AVERAGE).
• Median: sort the data, take the middle; for an even count, average the two middle values (MEDIAN).
• Mode: the most frequent value (MODE).

Spread: how scattered the data is.

• Quartiles cut sorted data into four equal parts. The first quartile is the median of the lower half, the second quartile is the overall median, the third quartile is the median of the upper half. A box-and-whisker plot draws the minimum, the three quartiles, and the maximum in one picture: the box runs from Q1 to Q3 and whiskers reach the extremes.
• Variance measures spread numerically: take each value's deviation (value − mean), square it, and average the squares. The bigger the variance, the more spread out the data.
• Standard deviation is the square root of the variance, back in the original unit, so it is easier to read than variance.

Picture it

flowchart LR
  D[Raw data] --> S[Sort it]
  S --> MN[Minimum]
  S --> Q1[First quartile Q1]
  S --> MED[Median Q2]
  S --> Q3[Third quartile Q3]
  S --> MX[Maximum]
  Q1 --> BOX[Box spans Q1 to Q3]
  Q3 --> BOX
  MN --> WL[Left whisker]
  MX --> WR[Right whisker]
  BOX --> PLOT[Box-and-whisker plot]
  WL --> PLOT
  WR --> PLOT

Worked example

Twelve game scores: 6, 9, 3, 6, 10, 12, 4, 3, 8, 10, 7, 6.

• Mean = (6+9+3+6+10+12+4+3+8+10+7+6) ÷ 12 = 84 ÷ 12 = 7 points.
• Median: sorted → 3, 3, 4, 6, 6, 6, 7, 8, 9, 10, 10, 12. Even count, so average the two middle values 6 and 7 → (6+7) ÷ 2 = 6.5 points.
• Mode: 6 appears most often → 6 points.

Now spread, with the six handball throws 26, 25, 32, 28, 32, 25 (mean = 28). The deviations are −2, −3, 4, 0, 4, −3; squared they are 4, 9, 16, 0, 16, 9, summing to 54. Variance = 54 ÷ 6 = 9, so standard deviation = √9 = 3 m.

Your turn

Practise computing mean, median, and mode, finding quartiles, reading a box-and-whisker plot, and comparing two datasets by their variance.

Recap

• Frequency tables and histograms organize data into classes.
• Centre: mean (AVERAGE), median (MEDIAN), mode (MODE).
• Quartiles Q1/Q2/Q3 split sorted data in four; a box plot shows min, Q1, median, Q3, max.
• Variance = mean of squared deviations; standard deviation = √variance, in the data's unit.