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O-Level Mathematics · Secondary 3 and 4

Statistics and Probability

Statistics is about summarising and comparing data, and probability is about measuring how likely events are. In O-Level Mathematics you calculate the mean, median, mode, quartiles, interquartile range and standard deviation, read cumulative frequency curves and box-and-whisker plots, and compare two sets of data. In probability you find the chance of single and combined events using possibility diagrams and tree diagrams. These skills are used whenever you interpret information from surveys, experiments and games of chance.

What the syllabus expects

  • Calculate the mean, median and mode from lists and frequency tables, and estimate the mean of grouped data using mid-values
  • Draw and interpret cumulative frequency curves to estimate the median, quartiles, percentiles and interquartile range
  • Interpret box-and-whisker plots
  • Calculate and interpret the standard deviation of a set of data
  • Compare two distributions using a measure of average and a measure of spread
  • Calculate probabilities of single events, and of combined events using possibility diagrams and tree diagrams
  • Add probabilities of mutually exclusive events and multiply probabilities of independent events

Averages and frequency tables

The mean is the total divided by the number of values, the median is the middle value when the data are in order, and the mode is the most common value. For a frequency table, the mean is Σfx ÷ Σf, where f is the frequency of each value x. For grouped data you do not know the exact values, so use the mid-value of each class as x, which gives an estimate of the mean. For the classes 0 < x ≤ 10, 10 < x ≤ 20 and 20 < x ≤ 30, the mid-values are 5, 15 and 25. To find the median from a frequency table of n values, find the position (n + 1)/2 and count through the cumulative frequencies. The mean uses every value and is affected by extreme values. The median is not affected by a few very large or very small values, which makes it a better average for skewed data such as incomes.

Cumulative frequency and box plots

A cumulative frequency curve shows how many values are less than or equal to each upper class boundary. Plot each cumulative frequency at the upper boundary of its class, not at the mid-value, and join the points with a smooth curve. For n values, read the median at n/2, the lower quartile at n/4 and the upper quartile at 3n/4 on the vertical axis, then read across to the curve and down to the horizontal axis. The interquartile range is the upper quartile minus the lower quartile and measures the spread of the middle half of the data. A box-and-whisker plot shows five values: the minimum, lower quartile, median, upper quartile and maximum. The box covers the middle 50% of the data. Percentiles work in the same way, so the 90th percentile is read at 90% of n.

Standard deviation and comparing data

Standard deviation measures how spread out the values are around the mean. A small standard deviation means the values are close to the mean and the data are consistent. For a frequency table, standard deviation = √(Σfx²/Σf − (Σfx/Σf)²). Your calculator can find it in statistics mode, but you should still be able to use the formula. When comparing two sets of data, make one comparison about the average and one about the spread, and write each in context. For example: 'Class A has a higher median score than Class B, so Class A generally scored higher. Class A has a smaller interquartile range, so its scores were more consistent.' Compare medians together with interquartile ranges, or means together with standard deviations.

Probability of combined events

The probability of an event is the number of favourable outcomes divided by the total number of equally likely outcomes, so it lies between 0 and 1. The probability that an event does not happen is 1 minus the probability that it does. For two events, a possibility diagram lists all the outcomes in a grid. With two fair dice there are 36 equally likely outcomes, and the sum is 7 in 6 of them, so P(sum of 7) = 6/36 = 1/6. A tree diagram shows events in stages. Multiply along the branches for 'and', since the stages happen one after the other, and add the results of separate paths for 'or', because those paths are mutually exclusive. If items are taken without replacement, the probabilities on the second set of branches change, because there is one fewer item in total.

Worked example

A bag contains 5 red balls and 3 blue balls. Two balls are taken at random, one after the other, without replacement. Find the probability that (a) both balls are red, (b) the balls are of different colours, (c) at least one ball is blue.

  1. First ball: P(red) = 5/8 and P(blue) = 3/8.
  2. Second ball after a red: 4 red and 3 blue remain, so P(red) = 4/7 and P(blue) = 3/7. After a blue: 5 red and 2 blue remain, so P(red) = 5/7 and P(blue) = 2/7.
  3. (a) P(red, red) = 5/8 × 4/7 = 20/56 = 5/14.
  4. (b) P(red, blue) + P(blue, red) = (5/8 × 3/7) + (3/8 × 5/7) = 15/56 + 15/56 = 30/56 = 15/28.
  5. (c) At least one blue is every outcome except red then red, so the probability is 1 − 5/14 = 9/14.
  6. Check: P(blue, blue) = 3/8 × 2/7 = 6/56, and 20/56 + 30/56 + 6/56 = 1, as it should be.

Answer: (a) 5/14 (b) 15/28 (c) 9/14

Common mistakes

  • Plotting cumulative frequency at mid-values. Each point must be at the upper class boundary, because cumulative frequency counts the values up to the end of the class.
  • Dividing by the number of classes instead of the total frequency. For a grouped mean, divide Σfx by Σf, the total number of values, not by the number of rows in the table.
  • Forgetting to change the second branches without replacement. After a red ball is taken from 5 red and 3 blue, there are 7 balls left, so the denominators become 7.
  • Adding when you should multiply. 'Red and then blue' needs multiplication along the branches. Adding is for combining separate outcomes, such as 'red then blue, or blue then red'.
  • Comparing data without context or without spread. 'Set A is higher' is incomplete. Say which average is higher, which spread is smaller, and what that means in the situation given.
  • Listing every path for 'at least one'. P(at least one blue) is quickest as 1 − P(no blue). Listing every path takes longer and makes it easy to miss one.

Exam tips

  • In cumulative frequency questions, draw clear horizontal and vertical lines on the graph to show where you read each value. This shows your method and helps you read the scale accurately.
  • In tree diagram questions, write the probabilities on every branch before answering any part, and check that each set of branches from one point adds up to 1.
  • Leave probabilities as fractions in their simplest form unless the question asks for a decimal. Fractions avoid rounding errors when you multiply along branches.

Practice questions

1. Find the mean, median, mode and range of 4, 7, 7, 9, 13.

Show answer

Mean = 40 ÷ 5 = 8. The median is the 3rd value, 7. The mode is 7. Range = 13 − 4 = 9.

2. In a test, 4 students scored marks in the class 0 < x ≤ 10, 10 students in 10 < x ≤ 20 and 6 students in 20 < x ≤ 30. Estimate the mean and the standard deviation.

Show answer

Mean 16 and standard deviation 7. Using mid-values 5, 15 and 25, Σfx = 20 + 150 + 150 = 320 and Σf = 20, so the mean is 320 ÷ 20 = 16. Σfx² = 4(25) + 10(225) + 6(625) = 6100, so the standard deviation = √(6100/20 − 16²) = √(305 − 256) = √49 = 7.

3. Two fair six-sided dice are thrown. Using a possibility diagram, find the probability that (a) the sum of the scores is 8, (b) at least one die shows a 6, (c) the product of the two scores is odd.

Show answer

(a) 5/36, from (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2). (b) 11/36, since 1 − P(no six) = 1 − 25/36. (c) 1/4. The product is odd only when both scores are odd, which happens in 3 × 3 = 9 of the 36 outcomes, and 9/36 = 1/4.

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