Summary — What This Chapter Covers
- Stem-and-leaf diagrams (including back-to-back) keep the raw data; always include a key.
- Box-and-whisker plots show the minimum, lower quartile, median, upper quartile and maximum.
- Histograms plot frequency density = frequency ÷ class width, so area represents frequency.
- Cumulative frequency graphs give estimates of the median, quartiles, percentiles and proportions above or below a value.
- Mean and standard deviation from raw data, grouped data, totals Σx and Σx2, or coded totals Σ(x − a) and Σ(x − a)2.
1. Stem-and-Leaf Diagrams, Box-and-Whisker Plots and Histograms
Diagrams
Choosing and drawing the right picture
A stem-and-leaf diagram keeps every data value and shows the shape; a back-to-back version compares two sets sharing the same stems (leaves on the left are read from right to left). It needs a key, e.g. "3 | 7 means 37", and for back-to-back a key for each side. It is unsuitable for large data sets.
A box-and-whisker plot shows the five-number summary and makes comparing medians and spreads easy, but hides the detail. A histogram is for grouped continuous data with possibly unequal classes: the height is the frequency density, so the area of each bar is proportional to its frequency. Use the true class boundaries (e.g. ages "10–14" in whole years have boundaries 10 and 15).
Histogram
frequency density = frequency / class width
Quartiles from raw data (n values in order)
Median = middle value. For the quartiles take the median of the lower and upper halves.
Worked Example 1
Grouped data: 0 ≤ x < 10 (frequency 8), 10 ≤ x < 15 (frequency 12), 15 ≤ x < 30 (frequency 9). Find the frequency densities for a histogram.
Worked Example 2
For the data 3, 5, 7, 8, 9, 11, 12, 14, 15, 18, 22 find the five-number summary for a box-and-whisker plot.
2. Cumulative Frequency Graphs
Cumulative frequency
Reading off medians, quartiles and percentiles
Plot cumulative frequency against the upper class boundary of each class and join with a smooth curve (or straight lines). For n values, read the median at n/2, the quartiles at n/4 and 3n/4, and the kth percentile at kn/100. To find how many values lie above a value x, read the cumulative frequency at x and subtract from n.
Without a graph, the same estimates come from linear interpolation within a class.
Worked Example
Heights of 100 plants (cm): cumulative frequencies are 10 at 150, 35 at 160, 70 at 170, 90 at 180, 100 at 190. Estimate the median, the interquartile range and the percentage of plants taller than 175 cm.
3. Measures of Central Tendency and Variation
Mean and SD
Totals, grouped data and coding
Mean, median and mode
range, IQR and standard deviation
For grouped data use mid-class values as x and frequencies f. With coded totals, the mean shifts by a but the standard deviation does not change.
Mean and variance
2
2
2
2
2
2
Coded data
2
2
2
Memorise