Secondary 3 Statistics Worksheet

8 of the 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 8 of the 12 questions are printed below. The other 4 are built on a diagram or a table of values that does not translate to the page, so they are in the free PDF — marked below where they would have come.

1. Q1Box and Whisker Plots

   This question is built around a diagram or a table of values. Open it in the PDF.

2. Q2Measures of Central Tendency

   Nine second-hand bicycles were sold at a neighbourhood swap. The prices paid, in dollars, were $$60,75,75,90,110,120,140,150,620.$$

   1. Calculate the mean price.
   2. Give the median price.
   3. Give the mode.
   4. Which of the three best describes what a buyer typically paid that day? Justify your choice.
3. Q3Measures of Dispersion

   This question is built around a diagram or a table of values. Open it in the PDF.

4. Q4Measures of Position

   Eleven swimmers were timed over $50$ m. Their times, in seconds, in order, were $$31,33,34,36,37,39,40,42,45,48,55.$$

   1. Give the minimum and the maximum of the distribution.
   2. Give the three quartiles.
   3. Zoé swam in $42$ s. Between which two of those values does her time sit, and what does that say about her place in the group?
   4. The quartiles are called measures of position rather than measures of dispersion. Explain the difference.
5. Q5Sampling Methods

   A regional sports federation has $4500$ registered players: $1800$ in soccer, $1500$ in hockey and $1200$ in basketball. It wants to question $150$ of them.

   1. The federation wants each sport represented in the same proportion as in the whole membership. How many players should be drawn from each sport? Show your work and check your three numbers.
   2. Name the sampling method used in a).
   3. Instead, the federation lists all $4500$ players, draws a random starting name among the first $30$, and then takes every $30$th name after it. Name that method and state how many players it selects.
6. Q6Scatter Plots

   This question is built around a diagram or a table of values. Open it in the PDF.

7. Q7Tables in Statistics

   Twenty students recorded how many times they used the city bike-share in one week: $$0,2,1,3,1,5,2,0,1,4,2,1,3,0,2,1,4,2,3,1.$$

   1. Build a table showing, for each value, the frequency and the relative frequency as a percentage.
   2. Check your table two ways.
   3. Give the mode and the range of the distribution.
   4. What percentage of the students used the bike-share at least three times?
8. Q8The Mean

   At a robotics competition, the $12$ members of team A scored a mean of $15$ points each, and the $8$ members of team B a mean of $20$ points each.

   1. How many points did each team score in total?
   2. Calculate the mean score of all $20$ competitors.
   3. Explain why the answer to b) is not $\frac{15+20}{2}=17.5$ points.
9. Q9The Median

   Ten pairs of second-hand skis were sold online. The prices, in dollars, were recorded in the order the sales happened: $$145,90,210,120,175,90,160,135,200,115.$$

   1. Write the ten prices in order.
   2. Find the median price.
   3. \'Etienne answers $132.50. Show exactly what he did, and say why it is wrong.
   4. Find the mean price and explain why it is not equal to the median here.
10. Q10The Mode

    This question is built around a diagram or a table of values. Open it in the PDF.

11. Q11The Quartiles

    Twelve students were asked how many minutes their trip to school takes. The answers, in the order given, were $$26,12,34,20,45,18,30,24,38,16,28,22.$$

    1. Write the twelve times in order.
    2. Find $Q_1$, $Q_2$ and $Q_3$.
    3. Calculate the interquartile range and the range.
    4. How many of the twelve students have a trip between $Q_1$ and $Q_3$? Say what the interquartile range means for this group.
12. Q12Synthesis — drawing on several sheets in this topic

    Eleven classes took part in a food drive. The amounts they raised, in dollars, were $$40,55,40,72,65,90,50,40,75,60,260.$$

    1. Find the mean, the median and the mode.
    2. Find $Q_1$, $Q_3$, the interquartile range and the range.
    3. Which single number best describes what a typical class raised? Explain why the mean is misleading here, supporting your answer with a count.
    4. Between which two amounts does the middle half of the classes lie, and what does comparing that width with the range show?

The 10 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The mean feels every value; the median only feels the middle

This one sentence explains Q2, Q9 and Q12, and it is the most useful idea on the sheet. The mean is computed from every number, so one unusual value pulls it. The median is a position in the ordered list, so an extreme value can be as extreme as it likes and the middle stays where it is.

Q2's bicycle sale makes it concrete. Eight ordinary bicycles and one vintage one at $620 give a mean of $160 — a price higher than eight of the nine bicycles actually sold. The median, $110, sits among the prices people really paid. Q12 repeats the shape with a food drive where one class raises $260, and asks you to support the choice with a count: nine of the eleven classes raised less than the mean. That count is the answer; "the mean is affected by outliers" on its own is not.

Sort the list. Every time. (Q9, Q11)

With an odd number of values the median is the single middle one. With an even number it is the mean of the two middle values, and it may well be a number that appears nowhere in the data — that is correct, not a mistake. Q11's twelve travel times land exactly there.

The quartiles follow the same rule one level down: Q₂ is the median, Q₁ is the median of the lower half and Q₃ the median of the upper half. Split the list at the median first, then treat each half as its own little distribution.

Position or dispersion? (Q4)

Q4 asks why the quartiles are called measures of position rather than measures of dispersion, and the distinction runs through the whole sheet. A measure of position names a place in the ordered distribution — a value that a stated fraction of the data falls below. A measure of dispersion describes how spread out the data are, and it is always a difference.

So the interquartile range is a measure of dispersion built out of two measures of position, which is why the two ideas are easy to confuse and worth separating in writing.

Why the range is the weaker of the two (Q3)

Q3 gives two bus routes with identical ranges and asks what the interquartile range adds. The answer is the argument for the whole measure: the range is decided by two values only, so a single very late morning makes a reliable route look as erratic as an unreliable one. The interquartile range throws away the top and bottom quarters and measures the middle half, which is where a route's normal behaviour lives.

Q12 closes the same loop from the other side: a middle half spanning $35 inside an overall range of $220 says that the group is tightly bunched and one class is exceptional. Comparing the two numbers is the interpretation the question wants — a range six times the interquartile range is describing an outlier, not a distribution.

Reading a box-and-whisker plot (Q1)

A box plot draws the five-number summary — minimum, Q₁, median, Q₃, maximum — and its most useful property is also the least intuitive.

The other side of the same coin: a box plot records five positions and nothing else. Two very different data sets can produce the same plot, so it cannot tell you the mean, and it cannot tell you how the values inside a section are arranged.

A scatter plot shows association, and association is not cause (Q6)

Q6 plots practice time against mistakes and asks for a description in two parts. The direction is negative here — more practice, fewer mistakes — and the strength is read from how tightly the points hug a single path. Both judgments are made from the plot, and both need a justification, not just a verdict.

Part (c) then estimates a value beyond the data collected, by continuing the pattern by eye, and asks why that is only an estimate. Three honest reasons: nobody in the sample practised that long, the pattern cannot continue below zero mistakes, and the number of mistakes depends on other things the plot never recorded. Saying so is the mark.

Sampling: proportional, systematic, and how a method quietly fails (Q5)

Q5 covers the two methods this year names. Stratified sampling divides the population into groups and draws from each in proportion to its size — work out the fraction the sample represents, apply it to every group, and check that the parts add back to the total. Systematic sampling lists the population, picks a random start inside the first interval, and then takes every nth name after it.

Systematic sampling is the one with a hidden failure mode, and it is worth knowing before it appears on an assessment: if the interval happens to match a pattern already in the list, every person selected occupies the same position in their group, and the sample stops representing anything. The repair is either to remove the pattern — shuffle the list, or draw names at random — or to keep the groups deliberately and sample at random inside each one.

A weighted mean is not the mean of the means (Q8)

Q8 gives two teams with different sizes and different means, and asks for the mean of everyone. Averaging the two means treats the groups as though they were the same size. They are not, so the larger group must count for more.

The reliable method is to go back to totals: multiply each mean by its own count to recover that group's total, add the totals, and divide by the combined count. The same manoeuvre run backwards is what makes several harder questions on this material tractable — a mean and a count together give you a total, and a total is a number you can do arithmetic with. Notice the check the question builds in: the combined mean must land between the two group means, and closer to the larger group's.

Frequency tables, and the mean of a table (Q7)

Q7 turns twenty raw answers into a table of frequencies and relative frequencies, and asks you to check it two ways: the frequencies must add to the number of individuals, and the percentages must add to 100 %. Both checks take one line and catch a miscount immediately.

Reading a median off a table works the same way. Add the frequencies from the top until you reach the middle position, and the value you are in when you get there is the median.

The mode is the only measure a qualitative variable has (Q10)

Q10 asks for favourite flavours, and part (b) asks why no mean and no median can be computed. The reason is worth stating precisely: a mean needs the values to be added and divided, and there is no such thing as one flavour plus another; a median needs them to be put in order from smallest to largest, and flavours have no such order. The mode only needs counting, so it survives.

Two limits on the mode, both worth remembering. It names the winner but not the margin — the most popular option can still be the choice of a minority. And a distribution may have two modes, or none at all, which is real information about the group rather than a problem to be smoothed away.

Getting the most out of it

Write the ordered list before anything else

Median, quartiles, minimum, maximum, range: every one of them is read off a sorted list, and the data are almost never given sorted. Rewrite the values in order as your first line, then count positions. It is one line of work that removes the single most common error on this sheet.

Say which measure you chose, and why, in a sentence

Several questions ask you to pick the measure that best describes a group. The choice alone earns little; the justification is the answer, and the strongest justification is a count — how many values fall below the mean, how far the extreme value sits from the rest.

Attach the unit and the context to every number

"An interquartile range of 13 minutes means the middle half of the students travel within a 13-minute band." A bare number rarely takes the full mark, and writing the sentence is also the fastest way to notice that you computed something other than what was asked.

Common questions