Statistics and Probability
Averages, data representation, probability, and sampling methods.
In this topic
Averages and Spread
Three types of average: - Mean = sum of all values ÷ number of values - Median = middle value when data is ordered - Mode = most frequent value
Measures of spread: - Range = highest value - lowest value - Interquartile range (IQR) = Q3 - Q1
Key Points
- Mean uses all the data but is affected by outliers
- Median is not affected by extreme values
- Mode is the only average for categorical data
- For grouped data, use midpoints to estimate the mean
Example Questions
3Find the mean, median, and mode of: 3, 5, 5, 7, 8, 9, 12
Mean = 49/7 = 7. Median = 7 (middle value). Mode = 5 (appears twice).
[3 marks]
2The mean of 5 numbers is 8. Four of the numbers are 5, 7, 9, 11. Find the fifth number.
Total = 5 × 8 = 40. Sum of four = 32. Fifth number = 40 - 32 = 8
[2 marks]
1Calculate the mean of the following set of numbers: 4, 8, 6, 5, 7.
Mean = (4 + 8 + 6 + 5 + 7) / 5 = 30 / 5 = 6.
[1 mark]
2Find the median of the following data set: 12, 15, 11, 14, 13.
First, arrange the numbers in order: 11, 12, 13, 14, 15. Median = 13 (middle number).
[2 marks]
1A survey of 5 students showed their ages as follows: 18, 19, 17, 20, 19. Calculate the mode of the ages.
The mode is 19, as it appears most frequently (twice).
[1 mark]
3A set of data has a mean of 10 and consists of 6 numbers. If one number is 12, what is the total of the other five numbers?
Total = Mean x Number of values = 10 x 6 = 60. Total of other five = 60 - 12 = 48.
[3 marks]
2The following marks were obtained by a student in 5 tests: 70, 80, 60, 75, 85. Calculate the range of the marks.
Range = Highest mark - Lowest mark = 85 - 60 = 25.
[2 marks]
4A class of students scored the following in a test: 56, 72, 68, 45, 85, 90, 38. Calculate the standard deviation of the scores.
First, calculate the mean (56+72+68+45+85+90+38)/7 = 63. Then find each deviation from the mean, square them, find the average of those squares, and take the square root. Standard deviation = √(variance).
[4 marks]
2Show that the mean of the following numbers 3, 5, 8, 7, 9 is equal to 6.4.
Mean = (3 + 5 + 8 + 7 + 9) / 5 = 32 / 5 = 6.4. Therefore, the mean is 6.4.
[2 marks]
5A school records the heights of students (in cm): 150, 160, 165, 155, 170, 180, 175. Calculate the interquartile range (IQR) and explain what it indicates about the data set.
First, order the heights: 150, 155, 160, 165, 170, 175, 180. Q1 (1st quartile) = 157.5, Q3 (3rd quartile) = 172.5. IQR = Q3 - Q1 = 172.5 - 157.5 = 15. The IQR indicates the range of the middle 50% of the data.
[5 marks]
Probability
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain).
P(event) = number of favourable outcomes / total number of outcomes
Key rules: - P(not A) = 1 - P(A) - For independent events: P(A and B) = P(A) × P(B) - For mutually exclusive events: P(A or B) = P(A) + P(B)
Key Points
- Probabilities always sum to 1 for all possible outcomes
- Tree diagrams: multiply along branches, add between branches
- Relative frequency = experimental probability
- Expected frequency = probability × number of trials
Example Questions
1A bag contains 3 red, 5 blue, and 2 green balls. One is picked at random. Find P(blue).
P(blue) = 5/10 = 1/2
[1 mark]
2Two dice are rolled. Find the probability that the total is 7.
Combinations giving 7: (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) = 6 out of 36. P = 6/36 = 1/6
[2 marks]
1A bag contains 3 red balls and 5 blue balls. What is the probability of randomly choosing a red ball from the bag?
The total number of balls is 3 + 5 = 8. The probability of choosing a red ball is 3/8.
[1 mark]
1A fair six-sided die is rolled. What is the probability of rolling an even number?
The even numbers on a die are 2, 4, and 6. There are 3 even numbers out of 6 total. Probability = 3/6 = 1/2.
[1 mark]
2A box contains 4 green marbles, 2 yellow marbles, and 6 purple marbles. If one marble is drawn at random, what is the probability that it is neither green nor yellow?
Total marbles = 4 + 2 + 6 = 12. Non-green and non-yellow marbles = 6 (purple). Probability = 6/12 = 1/2.
[2 marks]
3In a class of 30 students, 18 students have a pet cat, and 12 students have a pet dog. If a student is chosen at random, what is the probability that this student has either a cat or a dog (or both)?
Let A be the students with cats (18) and B those with dogs (12). Assuming no overlap, P(A or B) = 18 + 12 / 30 = 30/30 = 1. However, if some have both, we need more info. Assume all 30 have either. The probability = 1.
[3 marks]
3A coin is flipped and a six-sided die is rolled. What is the probability of getting heads and an odd number?
Probability of heads = 1/2. Probability of an odd number on a die (1, 3, 5) = 3/6 = 1/2. Combined probability = 1/2 * 1/2 = 1/4.
[3 marks]
4A jar contains 10 red, 15 blue, and 5 green jelly beans. You select two jelly beans at random without replacement. What is the probability that both jelly beans are red?
Probability of first red = 10/30. After one red is taken, there are 9 red and 29 total left. Probability of second red = 9/29. Combined probability = (10/30) * (9/29) = 90/870 = 1/9.67.
[4 marks]
4A card is drawn from a standard deck of 52 playing cards. Calculate the probability of drawing a heart or a queen. Show your workings.
There are 13 hearts and 4 queens. However, one queen is a heart, so total = 13 + 4 - 1 = 16. Probability = 16/52 = 4/13.
[4 marks]
5A box contains 5 red, 3 green, and 2 yellow balls. You pick 2 balls at random without replacement. Calculate the probability that both balls are the same color. Show all working.
Total ways to pick 2 from 10 = 10C2 = 45. For red: 5C2 = 10; for green: 3C2 = 3; for yellow: 2C2 = 1. Total same color = 10 + 3 + 1 = 14. Probability = 14/45.
[5 marks]