Mathematics 4024 ยท Syllabus tracker
Statistics
Averages and range, frequency tables, grouped data, charts, histograms, cumulative frequency, quartiles and scatter diagrams.
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Quick revision
Statistics on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Averages and spread, grouped data, charts, histograms, cumulative frequency and scatter diagrams.
1Averages and range
- Mean = total รท number of values. Median = middle value in order (average the middle two if there is an even count).
- Mode = most common value. Range = largest โ smallest (a measure of spread).
- From a frequency table: mean = ฮฃfx รท ฮฃf; median position = (n + 1) รท 2.
- Missing value from a mean: total = mean ร number of values.
Formulas
Worked example
Scores 1, 2 and 3 have frequencies 4, 5 and 1. Find the mean and the median.
- ฮฃfx = 4 + 10 + 3 = 17; ฮฃf = 10.
- Mean = 1.7.
- Median: position 5.5 โ the 5th and 6th values are both 2.
Mean 1.7, median 2
2Grouped data
- You don't know the exact values, so use each class's midpoint: the mean is an estimate.
- The modal class is the class with the highest frequency (for unequal widths, the highest frequency density).
- The class containing the median: find the (n/2)th value by adding up frequencies.
Worked example
Heights 150โ160 cm (f = 4), 160โ170 (f = 10), 170โ180 (f = 6). Estimate the mean.
- Midpoints 155, 165, 175.
- ฮฃfx = 620 + 1650 + 1050 = 3320; ฮฃf = 20.
Estimated mean 166 cm
3Charts and diagrams
- Bar charts (gaps, equal widths), pictograms (with a key), pie charts (angle = frequency รท total ร 360ยฐ).
- Stem-and-leaf: always give a key (3 | 4 = 34); leaves in order; read the median by counting.
- Compare two distributions with an average AND a measure of spread, in context.
Worked example
A stem-and-leaf diagram has key 3 | 4 = 34. What does 5 | 1 mean?
- Stem = tens, leaf = units.
51
4Histograms
- With unequal class widths the vertical axis is frequency density = frequency รท class width.
- The area of each bar is the frequency.
- To read a frequency: frequency density ร class width. For part of a class, take that fraction of the bar.
Formulas
Worked example
A bar for 10 < x โฉฝ 25 has frequency density 2. A bar for 25 < x โฉฝ 30 has frequency 15. Find the first frequency and the second density.
- Width 15: 2 ร 15 = 30.
- Width 5: 15 รท 5 = 3.
Frequency 30; density 3
5Cumulative frequency, quartiles and IQR
- Cumulative frequency is a running total. Plot each total at the upper end of its class and join with a smooth curve.
- Read the median at n/2, the lower quartile at n/4, the upper quartile at 3n/4 (on the cumulative frequency axis).
- Interquartile range = UQ โ LQ: the spread of the middle half, not affected by extremes.
- 'How many more than x?' โ read the cumulative frequency at x and subtract from the total.
How to do it
- Make a running total column.
- Plot (upper class boundary, running total).
- Join with a smooth curve.
- Read across from n/4, n/2, 3n/4 and down to the x-axis.
Worked example
80 students: the graph gives LQ = 31 marks and UQ = 46 marks. Find the IQR. 52 students scored 40 or less: how many scored more than 40?
- IQR = 46 โ 31 = 15.
- 80 โ 52 = 28.
IQR 15 marks; 28 students
6Scatter diagrams
- Correlation: positive (both increase), negative (one up, one down), or none.
- Line of best fit: one ruled line following the trend, with points roughly balanced either side; it need not pass through (0, 0).
- Use it to estimate within the data (interpolation). Estimates outside the data (extrapolation) are unreliable.
- An outlier is a point that doesn't fit the pattern.
Worked example
As temperature rises, sales of hot drinks fall. Describe the correlation.
- One goes up while the other goes down.
Negative correlation
โCommon mistakes
| Donโt write | Write instead |
|---|---|
| Median of 3, 4, 5, 7, 9, 12 is 5 | Even count: average the middle two, 6 |
| Giving the frequency as the modal class | Give the class, e.g. 10โ20 |
| Plotting cumulative frequency at midpoints | Plot at the upper end of each class |
| Histogram bar height = frequency with unequal widths | Height = frequency density |
| Dividing ฮฃfx by the number of classes | Divide by ฮฃf, the total frequency |
โ In the exam
- Show ฮฃfx and ฮฃf in a table โ they earn method marks.
- When comparing data, compare an average AND a spread, and say what it means in context.
- Read graphs carefully โ check the scale on each axis first.