Secondary 1 Statistics Worksheet

10 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. 10 of the 11 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Measures of Dispersion

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

2. Q2Measures of Position

   Seven sunflowers in a community garden were measured, in centimetres: $$145,118,132,151,118,139,127.$$

   1. Write the seven heights in order, from shortest to tallest.
   2. Give the minimum and the maximum of the distribution.
   3. Marie-\`Eve's sunflower measures $139$ cm. Counting from the shortest, what position does it occupy?
   4. Explain why the data must be put in order before part c) can be answered.
3. Q3Methods Used to Collect Data

   For each situation, state which data collection method was used — a census, a poll (survey) or a study — and justify your choice in one sentence.

   1. A high school records the transport used by every one of its $640$ students to find how many take the bus.
   2. A biologist sits beside a bike path and notes the colour of every helmet that passes between 8:00 and 9:00.
   3. A radio station telephones $300$ of its listeners to ask them their favourite kind of music.
4. Q4Range

   A weather station recorded these midday temperatures during one week in January: $$-6^{\circ }\text{C},-2^{\circ }\text{C},1^{\circ }\text{C},-9^{\circ }\text{C},3^{\circ }\text{C},0^{\circ }\text{C},-4^{\circ }\text{C}.$$

   1. Calculate the range of these seven readings and say what it means here.
   2. The following Monday the station recorded $-{11}^{\circ }\text{C}$. By how much does the range of the eight readings change?
5. Q5Sampling Methods

   A school of $480$ students wants a sample of $60$ students.

   1. The $480$ names are written on identical slips, mixed in a box, and $60$ slips are drawn. Name the sampling method used.
   2. The alphabetical list of the $480$ students is used, a starting name is drawn at random, and then every $8$th name after it is taken. Name the sampling method used, and show where the number $8$ comes from.
   3. State the population and the sample in b).
   4. A third plan is to question the first $60$ students who walk through the door on Monday morning. Explain why this plan does not give a representative sample.
6. Q6Sources of Bias

   Each of these three survey plans contains one source of bias. Name the source in each case, then rewrite question a) so that it is neutral.

   1. “Don't you agree that our embarrassing, run-down skatepark should finally be rebuilt?”
   2. To find out what students think of the cafeteria food, a reporter questions only the students standing in the cafeteria line.
   3. Students are asked what they think of their gym teacher, while the gym teacher stands beside the interviewer.
7. Q7Tables in Statistics

   Twenty-four students were asked how many brothers and sisters they have: $$0,1,2,1,3,0,1,2,4,1,0,2,1,1,3,2,0,1,2,5,\backslash 1,0,2,1.$$

   1. Build a frequency table for this data.
   2. Check your table by adding the frequencies.
   3. Give the range of the distribution.
8. Q8The Mean

   A class collected returnable bottles for seven days: $$12,9,15,8,11,14,15.$$

   1. Calculate the mean number of bottles collected per day.
   2. On how many days did the class collect more than the mean?
   3. If the class kept up this mean, how many bottles would it collect in $30$ days?
9. Q9Types of Graphs in Statistics

   For each situation, choose the most suitable graph — a bar graph, a broken-line graph or a circle graph — and justify your choice in one sentence.

   1. Showing how the $24$ students of a class split among the four workshops offered, so that each workshop's share of the class is visible.
   2. Showing the temperature inside a greenhouse, recorded every hour from 6:00 to 18:00.
   3. Comparing the number of medals won by five different clubs at the same tournament.
10. Q10Types of Statistical Variables

    Say whether each variable is qualitative or quantitative, and for the quantitative ones say whether they are discrete or continuous.

    1. The brand of skates a player wears.
    2. The number of goals scored in a game.
    3. The mass, in kilograms, of a student's backpack.
    4. The T-shirt size ordered (S, M, L, XL).
    5. The time, in seconds, taken to run $100$ m.
11. Q11Synthesis — drawing on several sheets in this topic

    The student council of a school of $600$ students wants to report the mean time students spend travelling to school. Two members stood at the front door from 7:50 to 8:00 one morning and questioned the first eight students who arrived. The times, in minutes, were $$10,25,15,20,5,30,12,23.$$

    1. Calculate the mean and the range of the eight times.
    2. These eight students were not chosen by any sampling method — they simply happened to be there. Give two reasons why the mean of these eight times is likely to be wrong for the whole school.
    3. The council decides to start again with a systematic sample of $40$ students taken from the complete list of the $600$. Say exactly what it must do, and show where your interval comes from.

The 9 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 says where, the range says how spread out (Q1, Q4, Q8)

Q1 is the whole idea in one table: two snow-clearing crews with the same mean and very different results. The mean is the total shared equally, so it tells you where the distribution sits — and by itself it cannot tell one steady crew from one wildly inconsistent one. The range, the largest value minus the smallest, is what separates them.

Q8 puts the mean into a context where the follow-up matters more than the calculation: how many days beat the mean, and what the mean predicts over a longer stretch. A mean is a rate — per day, per student — so extending it to thirty days is a multiplication, and it is a prediction rather than a fact. Say which it is.

Census, poll or study — three ways of getting data (Q3)

Q3 asks which method each situation used, and asks for the justification in a sentence. The three are told apart by asking who or what was observed:

The distinction is not academic: it decides what can go wrong. A census cannot be unrepresentative but is often impossible to run. A poll depends entirely on who was asked and how the question was worded. A study avoids what people say about themselves and is limited to what can be seen from outside.

Population, sample, and two ways to draw one (Q5)

The population is everyone the conclusion is about; the sample is the part of it actually examined. Q5 asks you to state both explicitly, because a conclusion is only as wide as its population and it is easy to slide from one to the other mid-sentence.

The other method in Q5 is the simple random draw: identical slips, mixed, drawn blind. What makes it random is not the mixing but the fact that every member has the same chance of being chosen. That is the test to apply to any proposed method, and it is what the last part of Q5 fails: questioning the first sixty students through the door on a Monday morning gives no chance at all to anyone who arrives late, and being early may well go with living nearby, taking a certain bus, or playing a sport before class. A sample is not representative because it is large; it is representative because of how it was drawn.

Sources of bias, and why a bigger sample does not fix them (Q6)

Q6 gives three survey plans, each spoiled in a different way, and asks you to name the source and then rewrite one question neutrally. The three are worth learning as separate faults, because they are repaired differently:

What kind of variable is it? (Q10)

Every graph choice and every calculation on the sheet depends on this classification, and Q10 is where it is made explicit.

The trap is that a variable can be written with numbers and still be qualitative — a jersey number or a postal code identifies rather than measures, and averaging one produces a number that means nothing. The test is not “does it look like a number?” but “would adding two of these values, or averaging them, mean anything?”

Frequency tables and the choice of graph (Q7, Q9)

A frequency table turns a list of raw answers into a count per value, and Q7 asks you to check it by adding the frequencies back to the number of people surveyed. That check is not ceremony: it is the only way to notice that one answer was skipped or counted twice while tallying, and it takes a second.

Q9 asks for the choice and the justification, and the justification is the mark. The two errors it is guarding against are joining up categories that have no order — a line drawn across four different sports claims a progression that does not exist — and putting into a circle a set of counts that do not divide a whole.

Q11: the synthesis question

The last free question runs the sheet end to end on one small data set: compute the mean and the range, criticise the sample the numbers came from, then design the sample that should have been taken. The eight students questioned at the door were not chosen by any method — they were simply present — so the criticism has to name specific reasons rather than say the sample is small. Then the repair is the systematic procedure above, with the interval shown as a division of the population size by the sample size, drawn from the complete list of the level.

That order — compute, then question the data, then propose a better method — is the shape of most statistics answers you will be asked for at this level, and it is worth practising as a sequence rather than as three separate skills.

Getting the most out of it

Write the data out in order before you touch it

Sorting costs one line and it hands you the minimum, the maximum and the range immediately, with any repeated value sitting visibly beside itself. Nearly every bookkeeping error on this topic — a value counted twice in a tally, a missing reading, a range subtracted the wrong way round — becomes obvious in a sorted list.

Interpret every number in a sentence, with its unit

“The range is 12” is half an answer; “the range is 12 °C, so the week's temperatures were spread over twelve degrees” is the whole one. Several questions ask explicitly what the number tells the supervisor, the coach or the council, and that sentence is the part being marked.

Ask who was left out

For every survey on this sheet, name a group of people the method could not reach. That single habit answers the sampling questions, the bias questions and the criticism part of the last question, because it is the same question underneath all three.

Common questions