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Statistics and Probability

Mathematics · WAEC and JAMB · SS2 and SS3

Statistics is the most predictable part of the paper and usually the easiest full question to score, built around a frequency table. Probability questions are short but trip candidates on the difference between drawing with and without replacement.

What you need to know

  • For ungrouped data, the mean is the sum of all values divided by how many there are, the median is the middle value after ARRANGING in order, and the mode is the value occurring most often.
  • With an even number of values, the median is the average of the two middle values, so for 4, 7, 9, 12 the median is (7 + 9)/2 = 8.
  • For a frequency table, the mean is the sum of fx divided by the sum of f. Build the fx column as a separate column in your table; the examiner awards marks for the table itself.
  • For grouped data, use the class MIDPOINT as x. The midpoint of the class 11-15 is (11 + 15)/2 = 13.
  • Class boundaries differ from class limits. For classes 1-5 and 6-10 the boundaries are 0.5-5.5 and 5.5-10.5, and histograms must be drawn with BOUNDARIES so that the bars touch.
  • The modal class is the class with the highest frequency, and the median class is the one containing the (N/2)th value when the cumulative frequency is built up.
  • The range is the largest value minus the smallest, which is quick but is distorted by a single extreme value.
  • Standard deviation measures the average spread about the mean: find each deviation from the mean, square them, average the squares to get the variance, and take the square root.
  • A pie chart turns each category into an angle: angle = (frequency/total frequency) x 360 degrees. All the angles must add to 360, and checking that sum catches arithmetic errors.
  • A cumulative frequency curve, or ogive, is plotted against the UPPER class boundary. From it you read the median at the N/2 level, the lower quartile at N/4 and the upper quartile at 3N/4.
  • The interquartile range is the upper quartile minus the lower quartile, and the semi-interquartile range is half of that. Read the question carefully for which one is wanted.
  • Probability of an event = number of favourable outcomes divided by total number of equally likely outcomes, and the answer always lies between 0 and 1. A probability greater than 1 means you have made an error.
  • For independent events, multiply: P(A and B) = P(A) x P(B). For mutually exclusive events, add: P(A or B) = P(A) + P(B). The word 'and' usually means multiply, 'or' usually means add.
  • When an item is drawn WITHOUT replacement, both the numerator and the denominator fall for the second draw: from 5 red in 12 balls, P(two reds) = (5/12) x (4/11), not (5/12) x (5/12).

Key terms

Mean
The sum of all the values divided by the number of values, also called the arithmetic average.
Median
The middle value when the data has been arranged in order of size.
Modal class
The class interval in a grouped frequency distribution with the highest frequency.
Class boundary
The value midway between the upper limit of one class and the lower limit of the next, used so that histogram bars touch.
Standard deviation
The square root of the mean of the squared deviations from the mean, measuring how widely the data is spread.
Cumulative frequency
The running total of frequencies up to and including the upper boundary of each class.
Mutually exclusive events
Two events that cannot both happen at the same time, so their probabilities may simply be added.

Formulae

  • mean of ungrouped data = (sum of x) / n
  • mean from a frequency table = (sum of f*x) / (sum of f)
  • class midpoint = (lower limit + upper limit) / 2
  • range = highest value - lowest value
  • variance = (sum of f*(x - mean)^2) / (sum of f)
  • standard deviation = sqrt(variance)
  • pie chart angle = (frequency / total frequency) x 360 degrees
  • interquartile range = Q3 - Q1; semi-interquartile range = (Q3 - Q1)/2
  • P(event) = number of favourable outcomes / total number of outcomes
  • P(A and B) for independent events = P(A) x P(B)
  • P(not A) = 1 - P(A)

Worked examples

The marks of 30 students are grouped as follows: 1-5 (4 students), 6-10 (7 students), 11-15 (10 students), 16-20 (6 students), 21-25 (3 students). Calculate the mean mark.

  1. Find the midpoint of each class: (1+5)/2 = 3, (6+10)/2 = 8, (11+15)/2 = 13, (16+20)/2 = 18, (21+25)/2 = 23.
  2. Multiply each midpoint by its frequency: 4 x 3 = 12, 7 x 8 = 56, 10 x 13 = 130, 6 x 18 = 108, 3 x 23 = 69.
  3. Add the fx column: 12 + 56 + 130 + 108 + 69 = 375.
  4. Add the frequencies: 4 + 7 + 10 + 6 + 3 = 30, which agrees with the 30 students stated.
  5. Mean = 375/30 = 12.5.

Find the standard deviation of the numbers 2, 4, 5, 6, 8.

  1. Find the mean: (2 + 4 + 5 + 6 + 8)/5 = 25/5 = 5.
  2. Find each deviation from the mean: -3, -1, 0, 1, 3.
  3. Square each deviation: 9, 1, 0, 1, 9.
  4. Add the squares: 9 + 1 + 0 + 1 + 9 = 20.
  5. Divide by the number of values to get the variance: 20/5 = 4.
  6. Standard deviation = sqrt(4) = 2.

A bag contains 5 red, 4 blue and 3 green balls. Two balls are drawn one after the other without replacement. Find the probability that both are red, and the probability that both are the same colour.

  1. Total number of balls = 5 + 4 + 3 = 12.
  2. P(first red) = 5/12. After removing one red, 4 reds remain out of 11 balls, so P(second red) = 4/11.
  3. P(both red) = (5/12) x (4/11) = 20/132 = 5/33.
  4. P(both blue) = (4/12) x (3/11) = 12/132.
  5. P(both green) = (3/12) x (2/11) = 6/132.
  6. Same colour means red or blue or green, so add: 20/132 + 12/132 + 6/132 = 38/132.
  7. Reduce: 38/132 = 19/66.

The mistake to avoid

In grouped data, candidates use the class limits or the frequency itself as x instead of the class midpoint, which makes every later figure wrong. In probability without replacement, they forget to reduce the denominator for the second draw, turning 4/11 into 4/12 and losing the whole answer.

In the exam

Set out your table with clear columns for x, f, fx and, where needed, cumulative frequency, because the completed table attracts marks even before the mean is found. Check that the frequencies add to the total the question gives you, and that pie chart angles add to 360. Leave probabilities as fractions in lowest terms unless the paper asks for decimals.