Year 10 and 10A Statistics: free practice test with answers

Sample statistics questions with answers

Try each one before you open the answer.

1. Collecting representative data

   A school has 1000 students: 400 in the junior school and 600 in the senior school. A stratified sample of 50 students is chosen in proportion to the size of each group.

   (a) How many junior students are in the sample?

   (b) How many senior students are in the sample?

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   Key idea: In a stratified sample, each group is sampled in proportion to its size.

   1. Junior: $\frac{400}{1000} \times 50 = 20$
   2. Senior: $\frac{600}{1000} \times 50 = 30$

   Answer: (a) $20$ (b) $30$

2. Review of data displays

   A stem-and-leaf plot shows the number of minutes 11 students spent on homework one night (stem = tens, leaf = units):

   Stem	Leaf
   1	2 5 8
   2	1 3 3 7
   3	0 4 8
   4	1

   Find the range, the mode, the median and the mean (correct to one decimal place).

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   Key idea: Read the data values from the plot in order, then calculate each statistic.

   1. Data in order: 12, 15, 18, 21, 23, 23, 27, 30, 34, 38, 41 (11 values).
   2. Range $= 41 - 12 = 29$ minutes.
   3. Mode: 23 appears twice, so the mode is 23 minutes.
   4. Median: the middle (6th) value is 23 minutes.
   5. Mean: sum $= 282$, so mean $= \frac{282}{11} = 25.636\ldots \approx 25.6$ minutes.

   Answer: Range 29 min, mode 23 min, median 23 min, mean 25.6 min

3. Two-way tables

   A survey of 80 students recorded whether each student plays a sport (Yes or No) and their year level (Year 9 or Year 10). The partially completed two-way table is shown below.

   	Plays sport	Does not play sport	Total
   Year 9	27	13	40
   Year 10	18	?	40
   Total	?	?	80

   Find the three missing values.

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   Key idea: Every row total and column total must add up to the grand total of 80.

   1. Find the missing Year 10 frequency for "Does not play sport":\

   $$40 - 18 = 22$$

   2. Find the column total for "Plays sport":\

   $$27 + 18 = 45$$

   3. Find the column total for "Does not play sport":\

   $$13 + 22 = 35$$

   Check: $45 + 35 = 80$ ✓

   Answer: The missing values are 22, 45 and 35.

4. Summary statistics

   Find the median, Q1, Q3 and interquartile range for the dataset: 12, 7, 15, 3, 9, 18, 11, 5, 14.

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   Key idea: Sort the data, find the median, then find Q1 and Q3 as the medians of the lower and upper halves.

   1. Sort the data: 3, 5, 7, 9, 11, 12, 14, 15, 18 (9 values).
   2. The median (Q2) is the 5th value: $Q2 = 11$.
   3. Lower half (values below the median): 3, 5, 7, 9. Q1 is the median of these four values: $Q1 = \frac{5 + 7}{2} = 6.$
   4. Upper half (values above the median): 12, 14, 15, 18. Q3 is the median of these four values: $Q3 = \frac{14 + 15}{2} = 14.5.$
   5. Calculate the IQR: $\text{IQR} = Q3 - Q1 = 14.5 - 6 = 8.5.$

   Answer: $Q2 = 11$, $Q1 = 6$, $Q3 = 14.5$, $\text{IQR} = 8.5$.

5. Standard deviation

   Calculate the mean and population standard deviation of the following five values: 4, 7, 7, 9, 8. Give the standard deviation correct to two decimal places.

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   Key idea: The population standard deviation is $\sigma = \sqrt{\dfrac{\sum(x - \bar{x})^2}{n}}$

   1. Find the mean: $\bar{x} = \frac{4+7+7+9+8}{5} = \frac{35}{5} = 7$

   2. Find each squared deviation from the mean:\

   $$\begin{aligned}(4-7)^2 &= 9\\(7-7)^2 &= 0\\(7-7)^2 &= 0\\(9-7)^2 &= 4\\(8-7)^2 &= 1\end{aligned}$$

   3. Find the population variance: $\sigma^2 = \frac{9+0+0+4+1}{5} = \frac{14}{5} = 2.8$

   4. Find the population standard deviation: $\sigma = \sqrt{2.8} \approx 1.67$

   Answer: Mean $= 7$, population standard deviation $\approx 1.67$

6. Cumulative frequency and percentiles

   The quiz scores of 20 students are summarised below.

   Score	5	6	7	8	9
   Frequency	2	5	8	4	1

   (a) Write the cumulative frequency for each score.

   (b) Find the median score.

   (c) What percentage of students scored 6 or less?

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   Key idea: Cumulative frequency is the running total of frequencies. It tells you how many values are less than or equal to each score.

   (a)

   1. $2$, $2 + 5 = 7$, $7 + 8 = 15$, $15 + 4 = 19$, $19 + 1 = 20$.

   (b)

   1. With 20 scores, the median is the average of the 10th and 11th scores.
   2. The cumulative frequency passes 7 at score 6 and reaches 15 at score 7, so the 10th and 11th scores are both 7.
   3. Median $= 7$.

   (c)

   1. Cumulative frequency at score 6 is 7.
   2. $\frac{7}{20} \times 100\% = 35\%$

   Answer: (a) 2, 7, 15, 19, 20 (b) 7 (c) 35%

7. Time-series data

   A café's coffee sales (cups per week) over five weeks were:

   Week	1	2	3	4	5
   Cups	400	420	450	470	500

   (a) By how many cups did weekly sales increase from Week 1 to Week 5?

   (b) What percentage increase is this?

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   Key idea: Percentage increase $= \frac{\text{increase}}{\text{original}} \times 100\%$.

   1. $500 - 400 = 100$ cups
   2. $\frac{100}{400} \times 100\% = 25\%$

   Answer: (a) $100$ (b) $25\%$

8. Bivariate data and scatter plots

   A scatterplot is drawn using the following eight data points, where $x$ is the number of hours a student studies per week and $y$ is their test score out of 100.

   Hours studied ($x$)	2	4	5	6	7	8	9	10
   Test score ($y$)	45	52	58	63	68	71	78	82

   Describe the association between hours studied and test score in terms of direction, form and strength.

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   Key idea: Association is described by direction (positive or negative), form (linear or non-linear) and strength (strong, moderate or weak).

   1. Direction: As hours studied increases, test score also increases, so the association is positive.
   2. Form: The points rise at a roughly constant rate with no obvious curve, so the form is linear.
   3. Strength: The points follow the linear trend closely with little scatter, so the association is strong.

   Answer: Strong positive linear association.

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 10 two-way tables worksheet
• Year 10 statistics worksheet
• Year 10A mean and standard deviation worksheet
• Year 10 scatterplots and lines of good fit worksheet