Secondary 2 Statistics Worksheet
Secondary 2 statistics is two questions asked over and over: what is the typical value here, and how spread out is it? The mean answers the first and the range answers the second — but this year they arrive attached to a frequency table, where every value carries a weight, and to distributions that have to be combined rather than simply averaged. Around them sits everything that decides whether the numbers deserve to be trusted at all: census against poll, proportional and systematic sampling, the three places bias hides, relative frequencies, variable types, and picking a graph that does not mislead. Read it on the page, or print the free PDF.
Practice worksheet — free PDF
No email, no account, no watermark. Teachers: photocopy it for your classes freely. Worked solutions and 9 harder problems come with the Secondary 2 Math bundle.
5 of the 11 questions
Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 5 of the 11 questions are printed below. The other 6 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.
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Q1Measures of Dispersion
This question is built around a diagram or a table of values. Open it in the PDF.
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Q2Measures of Position
Nine riders took part in a skatepark contest. Their scores, in the order the riders competed, were
- Give the minimum and the maximum score, and the range.
- Kenza scored . What position does she occupy counting from the highest score? And counting from the lowest?
- Add your two answers to b). Explain why that total is for every rider whose score no one else obtained.
- Two riders both scored . What position do they occupy, and what position does the rider who scored then occupy?
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Q3Methods Used to Collect Data
For each investigation, name the data collection method used — a census, a poll (survey) or a study — and state what the population is.
- A maple-syrup producer weighs every one of the buckets emptied on Tuesday, to find the total mass collected that day.
- A city telephones of its households to ask whether they would use a bike path along Rue Sainte-Anne.
- A biology class counts the maple keys lying in each of squares of chosen at random in a park, to estimate how many the park holds.
For the investigation that used a poll, name the sample and give its size.
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Q4Range
This question is built around a diagram or a table of values. Open it in the PDF.
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Q5Sampling Methods
A festival has volunteers: in setup, in the kitchen and in security. The organizers want a sample of volunteers in which each team keeps its share of the whole.
- Name this sampling method, and find how many volunteers must be drawn from each team. Check your three numbers.
- Describe precisely how the setup volunteers in the sample should be chosen.
- Instead, the organizers take the alphabetical list of the volunteers, draw a starting name at random and then take every th name after it. Name that method and show that it also produces volunteers.
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Q6Sources of Bias
Each survey plan below contains one source of bias. Name the source in each case — the choice of the sample, the wording of the question or the conditions of the interview — and justify it in one sentence.
- To find out how often the people of a neighbourhood use the public pool, an interviewer questions people as they leave the pool on Saturday afternoon.
- “Do you support the new bike path, which will make the street safer for children?”
- A questionnaire about how much money students spend at lunch must be signed with the student's name and handed back to the class teacher.
Then rewrite question b) so that it is neutral.
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Q7Tables in Statistics
This question is built around a diagram or a table of values. Open it in the PDF.
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Q8The Mean
This question is built around a diagram or a table of values. Open it in the PDF.
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Q9Types of Graphs in Statistics
This question is built around a diagram or a table of values. Open it in the PDF.
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Q10Types of Statistical Variables
A ski club collects the following information about each of its members: the number of runs completed in a day, the time spent on the slopes (in hours), the type of pass (day, season or student), and the region the member comes from.
- For each of the four variables, say whether it is qualitative or quantitative, and for the quantitative ones whether they are discrete or continuous.
- Give a plausible value for each variable.
- Which of the four variables can take the value , and which cannot? Explain.
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Q11Synthesis — drawing on several sheets in this topic
This question is built around a diagram or a table of values. Open it in the PDF.
The 9 challenge problems for this topic are a separate, paid sheet and are not reproduced here.
Which stream is this for? There is no stream to choose at this level. Secondary 2 is the second year of Cycle One and every student follows the same program, so this sheet is simply Secondary 2 mathematics as the Québec Education Program's Progression of Learning sets it out — the streams begin in Secondary 4.
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.
Two measures, and what each one is blind to
The mean and the range are the two numbers this level works with, and almost every question uses one of them. They are worth defining against each other: the mean is the total shared out equally among all the data, and the range is the largest value minus the smallest.
mean = sum of all the values ⁄ number of values range = maximum − minimumThe range uses two values and ignores every other one. Q1 is built on exactly that: one week has five of its seven days within two customers of each other and still has the largest range of the three, because a single low day and a single high day decide the number by themselves. The range says how wide the distribution is, and nothing whatever about how the data sit inside that width.
The mean uses every value and hides where they are. Two very different distributions can share a mean exactly. That is why Q11 asks for the mean, the range, and then how many days were above the mean — three questions, because no one of them describes the distribution on its own.
The mean of a frequency table is a weighted mean (Q8)
This is the most important idea on the sheet, and Q8 makes the error visible before explaining it. Forty students report how many books they read; the table lists six possible values, 0 through 5, and how many students gave each. Averaging the six values — (0+1+2+3+4+5)⁄6 — gives 2.5 and is wrong, because it treats a value chosen by ten students as counting the same as a value chosen by three.
mean = Σ(value × its frequency) ⁄ Σ(frequencies)The top row of the table is the list of possible values; the bottom row is how much each one weighs. Multiply across, add the products, and divide by the total number of individuals — not by the number of columns. Checking that the frequencies really do add to the stated total, which Q8 and Q11 both ask for first, is what stops a wrong denominator from going unnoticed.
Means of groups do not average
Take two classes, one of twelve students averaging fifteen correct answers and one of eighteen averaging twenty, and report (15 + 20)⁄2 = 17.5 for the thirty. It is the same mistake as Q8 in a new costume: the two means are being weighted equally when the groups are not the same size.
Go back through the totals. A mean is a total divided by a count, so recover each total — 12 × 15 and 18 × 20 — add them, and divide by 30. The combined mean is pulled towards the larger group, which is why it comes out above 17.5 here.
The simple average is right in exactly one case: when the two groups are the same size. Saying so is part of a complete answer, because it identifies what the shortcut was actually assuming.
Reading a range off a table without reading the wrong row (Q4)
A frequency table has two rows, and the range belongs to the row of values. Q4 shows a student computing 8 − 1 = 7 from the row of frequencies. That number is real — it is the spread of the counts, the difference between the most and least common results — but it is not the range of the distribution, and being able to say what it does measure is what the question wants.
Adding one more value changes the range only if it lands outside it. A thirty-first game with four goals sits inside the existing spread, so the minimum and maximum are untouched and the range does not move. The mean, by contrast, moves for every value added.
Shifting a whole distribution leaves the range alone. Add the same number to every value and both the maximum and the minimum rise by it, so the difference is unchanged — while the mean rises by that number exactly. Checking that on three sets at once is enough to see it is a rule and not a coincidence.
Position by rank (Q2)
Nine contest scores, listed in the order the riders competed. The first thing to do is order them; nothing about position can be read from the order of competing.
Q2 then notices something tidy: a rider's position counting from the top plus their position counting from the bottom always adds to ten. With nine riders that total is 9 + 1, because counting inwards from both ends counts the rider themselves twice. The result generalises to n + 1 — for any distribution where no two values are equal.
Ties break the pattern, and that is the interesting part. Two riders sharing a score share a position, and the next rider down does not simply take the following number. Decide how you are handling ties, write it down, and stay consistent — the arithmetic here is trivial and the bookkeeping is where the answer is won or lost.
How the data was collected, before any of it is trusted (Q3)
Q3 sorts three investigations and asks for the population each time. The three methods are worth separating precisely, because the words get used loosely in conversation:
Census — every individual of the population is examined. Weighing all eighty-four buckets emptied that day is a census, and its population is those eighty-four buckets.
Poll (survey) — a sample is questioned and the conclusion is extended to the whole population. Four hundred households out of twenty-one thousand: the population is the twenty-one thousand, the sample is the four hundred.
Study — observing and measuring rather than asking, often to estimate a quantity that cannot be counted directly, such as the maple keys in a park from thirty squares of one square metre.
The population is what the conclusion is about, not what was measured. Mixing the two up is what turns a fair survey into an overclaim, and it is the first thing to fix in any answer.
A census is exact and often impossible in practice, and costing out the phone calls needed to reach fifteen hundred members makes that concrete quickly. Cost, time and feasibility are the reasons samples exist — worth saying explicitly, because "just ask everyone" is otherwise the obvious answer to every sampling question.
Sampling methods: proportional stratified and systematic (Q5)
Q5 has a festival with three teams of different sizes and wants a sample in which each team keeps its share of the whole. That is a proportional stratified sample, and it is computed one stratum at a time:
number drawn from a group = (size of that group ⁄ size of the population) × sample sizeThen check the three answers add to the sample size. That check is what catches a sample built the lazy way: taking twenty students from each of three unequal school levels produces a sample of sixty that is not proportional at all. The over-represented level is the small one, and naming it is the point.
Within a stratum the choice must still be random. "Proportional" fixes how many come from each group, not which ones — Q5 asks how the setup volunteers should actually be picked, and the answer is a genuine random draw from that team's list, not the first names on it.
A systematic sample takes a random starting point in an ordered list and then every kth individual after it. Dividing the population by the sample size gives that step, and it lands on the same sample size by construction — which is worth verifying rather than asserting.
Bias has three homes, and a fourth that hides (Q6)
Q6 gives three flawed survey plans and asks which kind of bias each one carries. Naming the source is the mark; "it's biased" on its own is not an answer.
The choice of the sample — interviewing people as they leave the pool tells you about pool users and nothing about the neighbourhood, because everyone who never goes had no chance of being asked.
The wording of the question — attaching a reason to it ("which will make the street safer for children") steers the answer. A neutral rewrite says what is proposed and stops, and Q6 asks for that rewrite.
The conditions of the interview — signing a questionnaire about money and handing it to your own teacher changes what people write. Anonymity is a condition, not a courtesy.
Non-response is the fourth, and the one that hides in the arithmetic: when only sixty of five hundred questionnaires come back, a percentage computed on the sixty describes the people who chose to reply, not the membership. The people with a complaint are the ones most likely to answer, so the figure is not merely uncertain — it leans in a predictable direction.
Relative frequencies, and the check they come with (Q7)
Q7 has a table with one frequency missing, and asks for a row of relative frequencies as percentages. Each one is that category's count divided by the total, times a hundred — and then the five percentages have to add to 100 %.
That sum is a real check, not a formality. A total of 103 % means either a percentage was computed on the wrong denominator or the missing frequency was found wrongly, and the check catches both. Rounding each percentage can leave a sum of 99.9 or 100.1, which is not the same kind of problem and should be said in a clause rather than quietly corrected.
A percentage of a small group is not a small number of people. Take 35 % of four hundred students against 60 % of one hundred, and the conclusion that the second group has more. The percentages point one way and the counts the other, because a percentage always discards the size of the group it came from. When two groups differ in size, convert back to counts before comparing anything.
Choosing the graph, and how a graph misleads (Q9)
Each graph type answers one kind of question, and the justification is what is being marked:
Circle graph — how a whole divides into parts. Each sector's central angle is that category's share of 360°, and the four angles must add to 360° — the same kind of check as the percentages.
Bar graph — comparing counts across categories directly. The bars stand apart, because the categories are not a continuous scale.
Broken-line graph — how one quantity changes over time, which is what makes it the right choice for tracking bus users from September to June and the wrong choice for comparing four means of transport.
An axis that does not start at zero exaggerates every difference. Bars for 200, 210, 220 and 240 drawn from a baseline of 190 leave the last bar five times the height of the first, and a poster can then claim five times the sales. The heights are in a ratio of 5 to 1; the data is in a ratio of 1.2 to 1. Compute both and the exaggeration is impossible to argue with.
Qualitative, discrete, continuous (Q10)
Q10 labels four variables from a ski club. The useful test is not what the values look like but what a value between two observations would mean.
Qualitative — a category or a label: the type of pass, the region. Numbers that name rather than measure are qualitative too.
Quantitative discrete — counted, so only whole values occur: runs completed in a day. Half a run is not a possible answer.
Quantitative continuous — measured, so any value in an interval is possible: time on the slopes. 3.5 hours is meaningful; 3.5 runs is not.
The precision of the recording is a separate matter from the variable. A greenhouse temperature written to the nearest whole degree still varies continuously — the whole numbers are the instrument, not the quantity. That single counterexample is the reason "all the answers are whole numbers" never settles the question.
Q11: the four ideas over one table
The synthesis question gives twenty days of canteen sales as a frequency table and runs the mean, the range, a count against the mean, and two graph choices over it. Verify the frequencies add to twenty before anything else — every later part divides by that total, so an error there propagates through the whole question and nothing downstream will contradict it.
Getting the most out of it
Rewrite a frequency table as a column of products
Value, frequency, and their product side by side. It costs one line, it makes the total number of individuals and the total of the quantity both visible at once, and it is what stops the mean from being taken over the wrong denominator.
Say what the number means, with its unit, in a sentence
A mean of 2.3 is "2.3 books per student over the summer", and a range of 6 is "6 customers between the busiest and quietest day". The sentence is usually part of the answer, and it is also the check that catches a number computed off the wrong row.
Do the sampling and bias questions carefully, not quickly
They have no arithmetic and they look like the easy ones, but they are marked on the justification rather than the verdict. Name the population, name the sample, name the source of bias, and give the one sentence saying why — that sentence is the answer.
Want the solutions, or something more challenging?
The worksheet, the questions above and every explanation on this page stay free permanently. Three more PDFs exist for this topic — the worked answer key, a harder problem set, and the answer key to that. They come with the Secondary 2 Math Solutions Bundle, which is what keeps the rest of the series free.
What else exists for Statistics
Three PDFs · 14 pages · all three are in the bundle below.
- Answer key — 3 pages. All 11 questions worked step by step, including the restrictions and the justifications. Not a list of final answers.
- Challenge problems — 8 pages, 9 problems. A separate sheet at exam-plus difficulty covering the same 10 concepts. Harder than anything on the free sheet.
- Challenge answer key — 3 pages. Every challenge problem worked to the same standard, with the checks shown.
- PDF, letter size, print-ready.
The one thing that's for sale
Every Secondary 2 Math topic — the complete Solutions Bundle
One download, one payment, the whole program. Every answer key and every challenge set for all 14 Secondary 2 Math worksheet sets — including this one.
- Worked solutions, not answer lists — every step written out
- Covers the whole year's program at this level
- Less than the price of one hour of tutoring — for the entire year's solutions
Taking Secondary 1 Math as well? The Secondary 1 Math bundle covers all 15 of its sets — 45 PDFs, 182 pages — on the same terms.
Taking Secondary 3 Math as well? The Secondary 3 Math bundle covers all 11 of its sets — 33 PDFs, 154 pages — on the same terms.
Taking Secondary 4 Math as well? The Secondary 4 Math bundle covers all 17 of its sets — 51 PDFs, 154 pages — on the same terms.
Taking Secondary 5 Math as well? The Secondary 5 Math bundle covers all 21 of its sets — 63 PDFs, 228 pages — on the same terms.
Taking CEGEP Calculus I as well? The CEGEP Calculus I bundle covers all 8 of its sets — 24 PDFs, 82 pages — on the same terms.
Taking CEGEP Calculus II as well? The CEGEP Calculus II bundle covers all 8 of its sets — 24 PDFs, 90 pages — on the same terms.
Taking CEGEP Linear Algebra as well? The CEGEP Linear Algebra bundle covers all 7 of its sets — 21 PDFs, 82 pages — on the same terms.
Taking AP Calculus AB as well? The AP Calculus AB bundle covers all 8 of its sets — 24 PDFs, 126 pages — on the same terms.
Common questions
Is this worksheet really free?
Yes — the questions are on this page to read, and the PDF downloads directly, no email and no account. The one paid item is optional: the complete Secondary 2 Solutions Bundle, which covers every set at this level.
How do I find the mean from a frequency table?
Multiply each value by its frequency, add those products to get the total of the whole quantity, then divide by the total number of individuals — not by the number of columns. Averaging the top row on its own treats a value chosen by ten people as counting the same as one chosen by three, which is the most common error in this topic.
Why can't I average two class means to get the mean of both classes?
Because the classes are different sizes, so their means do not carry equal weight. Recover each class's total by multiplying its mean by its size, add the two totals, then divide by the combined number of students. The result leans towards the bigger class. Averaging the two means directly is correct only when the groups are the same size.
What exactly does the range tell me?
How wide the distribution is, and nothing else. It is the largest value minus the smallest, so it uses two values and ignores every other one — a distribution can be tightly bunched with a single stray value and still have a large range. Adding the same number to every value leaves the range unchanged while the mean shifts by that number.
Are the median, the mode and quartiles on this sheet?
No. At Secondary 2 the measures in play are the mean and the range, and this sheet stays inside that. The median, the mode, the quartiles, box-and-whisker plots and scatter plots all arrive in Secondary 3, so nothing here needs them.
What is the difference between a census, a poll and a study?
A census examines every individual of the population. A poll questions a sample and extends the conclusion to the whole population. A study observes or measures — often to estimate something that cannot be counted directly. In every case the population is what the conclusion is about, which is not always what was measured.
Which year is this sheet for?
Secondary 2, the second year of Cycle One, and it stays inside that year. Everything here is the mean, the range, and the collection and presentation of data as the Progression of Learning sets them out for this level.
Can teachers use this in class?
Yes. Print and photocopy it for your own classes freely — I just ask that the tutorinmontreal.ca footer stays on the page.
I'm stuck on one question. Can you help?
Yes — through one-on-one tutoring, in Montreal or online. Get in touch to arrange a session, or see the current rates.
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The same topic at the other level: Secondary 3 Math · Statistics.