Statistics, Probability and Data Depth
Statistics, Probability and Data Depth combines the harder end of Statistics and Probability with Probability and Data Handling into multi-step questions, including comparing data sets, estimating a mean from grouped data, and combined probability using tree diagrams and scatter graphs.
Before you start
Make sure you're comfortable with these topics first:
Method
- For 'compare two distributions' questions, give two comparisons: one about average (mean, median or mode) and one about spread (range, or consistency), since a single comparison rarely earns full marks.
- When a probability question is built on top of data from a table or list, first identify the total number of possible outcomes accurately before finding the probability of the specific event asked about.
- For 'without replacement' probability questions, remember the total number of items decreases after each pick, and multiply the probabilities along each branch of a tree diagram.
- For grouped frequency data, use the midpoint of each class interval to estimate the mean, and remember this gives an estimate, not an exact value, since the exact data inside each group is unknown.
- For scatter graphs, describe the type of correlation shown (positive, negative or none), and remember that correlation shows an association, not that one thing causes the other.
- Check that probabilities for all possible outcomes of a single event sum to 1, since this is a quick way to check a set of calculated probabilities is correct.
- Label every branch, row or cell in a tree diagram or table clearly, since depth-level data questions are usually multi-step and each labelled step can earn its own mark.
Worked example
A bag contains 5 red counters and 3 blue counters only. Yusuf picks two counters at random from the bag, one after the other, without putting the first counter back. Work out the probability that both counters are red.
- Find the probability the first counter is red: 5 out of 8 counters are red, so P(first red) = 5/8.
- Since the first counter is not replaced, 7 counters remain, with 4 red remaining, so P(second red, given the first was red) = 4/7.
- Multiply the two probabilities along the branch, since both events must happen: 5/8 x 4/7 = 20/56.
- Simplify the fraction. Final answer: 20/56 = 5/14.
Practice questions
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Q1Two hockey teams recorded the number of goals they scored in their last 5 matches. Team A: 2, 3, 2, 4, 4. Team B: 0, 1, 5, 6, 3. Compare the two teams by working out the mean and the range for each.Show answer
Answer: Both teams have the same mean, 3 goals (15 / 5 = 3 for each team), but Team A is more consistent, with a range of 2 (4 - 2), compared with Team B's range of 6 (6 - 0)
Q2A class recorded how many books each of 25 students read in a month: 5 students read 0-2 books, 12 students read 3-5 books, and 8 students read 6-8 books. Using midpoints 1, 4 and 7, estimate the mean number of books read.Show answer
Answer: 4.36 (fx values: 1x5=5, 4x12=48, 7x8=56, total = 109, mean = 109 / 25 = 4.36)
Q3A drawer contains 4 black socks and 6 white socks only. Two socks are taken out at random, one after another, without replacement. Work out the probability that both socks are white.Show answer
Answer: 1/3 (6/10 x 5/9 = 30/90 = 1/3)
Q4A fair coin is flipped and a fair six-sided dice is rolled at the same time. Work out the probability of getting heads on the coin and a number greater than 4 on the dice.Show answer
Answer: 1/6 (P(heads) = 1/2, P(greater than 4) = 2/6 = 1/3, combined = 1/2 x 1/3 = 1/6)
Q5A scatter graph plots the number of hours 10 students spent revising against their test scores. The points show that more hours of revision is generally associated with a higher test score, though not perfectly. Describe the type of correlation shown, and state one thing this correlation does not prove.Show answer
Answer: Positive correlation; it does not prove that revision caused the higher scores, since correlation does not imply causation, only an association between the two variables
Q6A spinner can land on red, blue, green or yellow only. P(red) = 0.3, P(blue) = 0.25, P(green) = 0.2. Work out P(yellow).Show answer
Answer: 0.25 (1 - (0.3 + 0.25 + 0.2) = 1 - 0.75 = 0.25)
Q7A researcher wants to find the average pocket money received by pupils at a school, so she surveys only the pupils waiting outside the head teacher's office. Give one reason this is likely to produce a biased sample.Show answer
Answer: Pupils outside the head teacher's office are not a fair, random cross-section of the whole school (they are a small group there for a specific reason), so the sample does not fairly represent every pupil
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
A box contains 7 milk chocolates and 5 dark chocolates only. Amara picks a chocolate at random, eats it, then picks a second chocolate at random from those remaining. (a) State the four probabilities that belong on the second set of branches of a probability tree diagram for this situation. (b) Work out the probability that Amara picks two chocolates of the same type.
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The mean of 6 numbers is 14. Five of the numbers are 9, 12, 18, 11 and 16. (a) Work out the sixth number. (b) A seventh number is then added to the list, and the mean of all 7 numbers becomes 15. Work out the seventh number.
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Free printable worksheet
Want more practice on paper? Download the statistics, probability and data depth worksheet pack - 20 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 22 of 13+ Maths Workbook, the whole course as one free printable PDF.
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