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Mathematics 4024

Quick revision notes ยท Mathematics 4024 ยท Chapter 7 of 35

Statistics

Averages and spread, grouped data, charts, histograms, cumulative frequency and scatter diagrams.

6 sectionsWorked examples Common mistakes
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At a glance

Averages and spread, grouped data, charts, histograms, cumulative frequency and scatter diagrams.

  1. Averages and range
  2. Grouped data
  3. Charts and diagrams
  4. Histograms
  5. Cumulative frequency, quartiles and IQR
  6. 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 โˆ’ 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

mean = ฮฃfx รท ฮฃffrequency table
median position = (n + 1) รท 2

Worked example

Scores 1, 2 and 3 have frequencies 4, 5 and 1. Find the mean and the median.

  1. ฮฃfx = 4 + 10 + 3 = 17; ฮฃf = 10.
  2. Mean = 1.7.
  3. 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 150โ€“160 cm (f = 4), 160โ€“170 (f = 10), 170โ€“180 (f = 6). Estimate the mean.

  1. Midpoints 155, 165, 175.
  2. 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?

  1. 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

frequency density = frequency รท class width
frequency = density ร— width

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.

  1. Width 15: 2 ร— 15 = 30.
  2. 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 โˆ’ 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

  1. Make a running total column.
  2. Plot (upper class boundary, running total).
  3. Join with a smooth curve.
  4. 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?

  1. IQR = 46 โˆ’ 31 = 15.
  2. 80 โˆ’ 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.

  1. One goes up while the other goes down.

Negative correlation

โœ—Common mistakes

Donโ€™t writeWrite 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. 10โ€“20
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.