92526 Chap.9 Histograms, Boxplots and Continuous Summaries
Histograms, Boxplots and Continuous Summaries
Continuous data is the more common kind in sport and needs extra steps before it can be pictured. Following the Week 3 slides, this chapter groups values into bins by computing the range, choosing a number of bins and dividing to get the bin width, then reads a histogram's shape: symmetric or skewed, bell-shaped or bimodal.
It sets out the centrality measures, mean, median and mode, and the variability measures, variance, interquartile range and range, and matches them to shape: mean with standard deviation for symmetric data, median with IQR for skewed data.
Boxplots follow from the five-number summary and fences at 1.5 times the IQR, worked through on a guard's points in eleven games.
The chapter closes with scatterplots for two continuous variables and line graphs for values over time, as built in the Week 4 tutorial.
What this chapter covers
- 01
Grouping continuous values into bins
- 02
Bin width from range and number of bins
- 03
Histogram shapes: symmetric, skewed, bimodal
- 04
Mean, median and mode
- 05
Variance, IQR and range
- 06
Five-number summary and boxplot fences
- 07
Scatterplots and line graphs
Worked example · free
Read a histogram of thirty jump heights
- 1Bin width = range divided by number of bins = (48 minus 24) divided by 6 = 4 cm, so the bins are 24 to 28, 28 to 32, up to 44 to 48.
- 1The tallest bar is the 32 to 36 cm bin with 9 athletes: 9 divided by 30 = 0.30, or 30%.
- 1At least 36 cm covers the last three bins: 8 + 4 + 2 = 14 athletes, and 14 divided by 30 = 46.7%.
- 1Shape: a single central peak with similar tails either side, roughly symmetric, so the mean and standard deviation are suitable summaries.
Key terms
- Histogram
- A chart of continuous data in which bars show how many values fall into each bin.
- Bin width
- The span of each histogram interval, found by dividing the range by the number of bins.
- Positive skew
- A distribution with a long tail to the right, which pulls the mean above the median.
- Five-number summary
- The minimum, lower quartile, median, upper quartile and maximum that a boxplot is built from.
- Scatterplot
- A graph plotting each observation at its pair of values for two continuous variables.
Histograms, Boxplots and Continuous Summaries FAQ
How do you choose the bins for a histogram?
Calculate the range of the data, decide how many bins you want, and divide the range by the number of bins to get the bin width. Excel will pick a width automatically, but the Week 4 tutorial shows it can be too wide to reveal detail, so adjust the bin width or number of bins until the shape is informative.
When should you report the median instead of the mean?
When the distribution is skewed or has outliers. A long tail drags the mean towards it while the median, the 50th percentile, barely moves. The slides pair the median with the interquartile range for left- or right-skewed data, and the mean with the standard deviation for symmetric, bell-shaped data.
How is an outlier identified on a boxplot?
Compute the interquartile range, Q3 minus Q1, then place fences 1.5 times the IQR below Q1 and above Q3. Any value beyond a fence is plotted as a separate point, and the whisker ends at the most extreme value still inside the fence rather than at the fence itself.
What does a boxplot show that a histogram does not?
A boxplot condenses a distribution into its five-number summary and flags outliers, which makes several groups easy to compare side by side. The Week 4 tutorial builds a grouped boxplot of scores by match result for exactly that purpose, while a histogram shows the detailed shape of one distribution.
When should points be joined with a line?
Only when the x-axis is time or another genuine sequence. The slides show values against successive matches first as a scatter and then joined into a line graph, because the line makes it easier to follow each value into the next. Joining unordered points invents a trend.
Exam move
Practise the three numerical routines until they are automatic: bin width from range and number of bins, the five-number summary with its fences, and the mean-versus-median check for skew. Use small invented datasets of ten or so values and work them by hand, stating which quartile method you use.
Then rebuild the Week 4 tutorial charts in Excel, a histogram with an adjusted bin width, a boxplot, a grouped boxplot and a scatterplot with axis titles, and write one sentence interpreting each, since explaining a chart to a coach is the skill the tutorials keep asking for.
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