Collect, organize and summarize data (Grade 8) — Mathematics Grades 8–11

Collecting and organising data, then finding the mean, median, mode and range.

Gr 8 Term 3 — Collect, organize and summarize dataTerm 3

Practise Collect, organize and summarize data in the app →

What gets asked

Judge whether a data question or a sample is a good one.
Organise raw data using tally marks or a frequency table.
Group data into equal, non-overlapping intervals.
Calculate the mean, median, mode or range of a data set.

You must be able to

Tell a sample apart from the whole population being studied.
Group tally marks in fives so counts are easy to check.
Order data before finding the median.
Divide by the number of values, not the number of categories, for the mean.

Traps that cost marks

Mixing up the sample and the population, calling the whole group the sample. The population is everyone the question is about. The sample is the smaller group actually asked.
Losing count while tallying because the marks are not grouped in fives. Group every five tally marks together. It makes the count much easier to check.
Finding the median before putting the data in order. Always order the data from smallest to largest first, then find the middle value.
Giving the smallest and largest values as the range. The range is one number: the largest value minus the smallest value.

Worked example

Five learners' airtime purchases in rands were:

10, 15, 15, 20, 30

. Find the mean and the mode.

Add the five amounts: $10 + 15 + 15 + 20 + 30 = 90$ .
Divide by how many values there are: $90 \div 5 = 18$ .
Mode is the value that appears most often: $15$ .
Mean is R18, mode is R15.

Collect and organise data

Who to ask, where to look it up, and how to count the answers without losing your place.

population: Every person or thing the question is about.
sample: A smaller group chosen from the population to speak for all of it.
source: Where the data comes from: the people you ask, or a book.
tally: One mark for each item as you count. The marks are grouped in fives.
frequency: How many times a value or a group turns up in the data.
interval: A group of values with a clear start and end, such as $0$ to $9$ .

First decide who the question is about. That whole group is the population. If you ask about learners at your school, the population is every learner there.

You can seldom ask everyone. You ask a sample and let it stand for the population. Choose it from every part of the group.

Match the source to the question. Your own class cannot tell you what the country thinks. Ask who made the data.

Now write the answers down. Make one tally mark for each item as you go. Group every five marks. Then count them and write the frequency beside each value.

Too many different values make a long table. Sort them into intervals instead, like $0$ to $9$ and $10$ to $19$ . Keep every interval the same width, and let no value fit two.

Examples

A Grade 8 group wants the favourite subject of all

840

learners at their school. Who is the population, and what is a fair sample?

Worked answer

The population is all $840$ learners at the school.
Asking all of them takes too long, so choose a sample.
Take some learners from every grade.
Asking only your friends leaves most of the school out.

Answer: population: all 840 learners; sample: some from every grade

The sample has to look like the population, or it cannot speak for it.

Twelve learners gave their shoe sizes:

5

;

6

;

5

;

7

;

6

;

5

;

8

;

6

;

5

;

7

;

6

;

6

. Make a frequency table.

Worked answer

Go down the list once, making a tally mark for each size.
Group the marks in fives so they stay easy to count.
Size $5$ has $4$ marks, size $6$ has $5$ , size $7$ has $2$ , size $8$ has $1$ .
Check the total: $4 + 5 + 2 + 1 = 12$ .

Answer: sizes 5, 6, 7 and 8 with frequencies 4, 5, 2 and 1

The frequency column must add up to the number of learners.

Learners' walking times to school run from

2

38

minutes. Sort them into intervals of

10

minutes.

Worked answer

Start at $0$ and step up in tens.
Use $0$ to $9$ , then $10$ to $19$ , then $20$ to $29$ , then $30$ to $39$ .
A time of $10$ minutes now has one home only, the second interval.
That is $4$ intervals, each $10$ minutes wide.

Answer: 0–9; 10–19; 20–29; 30–39 minutes

Equal intervals keep the picture fair.

Traps

Mixing up the sample and the population, calling the whole group the sample. The population is everyone the question is about. The sample is the smaller group actually asked.
Losing count while tallying because the marks are not grouped in fives. Group every five tally marks together. It makes the count much easier to check.
Choosing intervals like $0$ to $10$ and $10$ to $20$ , so a value of $10$ fits both. Give each interval a clear start and end: $0$ to $9$ , then $10$ to $19$ .
Picking a source that cannot reach the people the question is about. Match the source to the question. One class cannot speak for the country.

Learn and practise “Collect and organise data” in the app →

Summarise a data set

Four numbers that stand in for a whole list, and a display that keeps every value.

mean: Add all the values, then divide by how many there are. This is the everyday average.
median: The middle value, once the numbers are in order from smallest to largest.
mode: The value that turns up most often. A set can have more than one, or none.
extremes: The smallest value and the largest value in the set.
range: The largest value minus the smallest. It is one number, not two.
stem-and-leaf: A way of writing numbers with the tens digit shared and each ones digit listed.

The mean, median and mode each give a middle for a set. For the mean, add every value and divide by how many there are. Ten learners give ten values, so divide by $10$ .

The median is the middle value, but only once the list is in order. With an even count, two values sit in the middle. Add those two and halve.

The mode is the value that turns up most often. It is the value, never the number of times it came up.

The extremes are the smallest and largest value. The range is one number: largest minus smallest. It says how spread out the data is.

A stem-and-leaf display keeps every value. The tens digit is the stem, written once. Each ones digit is a leaf, written beside its stem. So $23$ , $25$ and $28$ give stem $2$ with leaves $3$ , $5$ , $8$ .

Rules to remember

mean $= \frac{sum of the values}{how many values}$
range $= largest - smallest$

Examples

Five learners bought airtime, in rands:

10

;

15

;

15

;

20

;

30

. Find the mean and the mode.

Worked answer

Add the five amounts: $10 + 15 + 15 + 20 + 30 = 90$ .
Divide by how many values there are: $90 \div 5 = 18$ .
$15$ comes up twice and nothing else repeats, so it is the mode.

Answer: mean R18, mode R15

Divide by $5$ , the number of learners, not by $4$ , the number of different amounts.

Six taxi fares in rands were

14

;

20

;

12

;

25

;

18

;

15

. Find the median and the range.

Worked answer

Put them in order: $12$ ; $14$ ; $15$ ; $18$ ; $20$ ; $25$ .
There are six fares, so two sit in the middle: $15$ and $18$ .
$\frac{15 + 18}{2} = 16, 5$ .
The extremes are $12$ and $25$ , so the range is $25 - 12 = 13$ .

Answer: median R16,50 and range R13

With an even count the median can be an amount nobody paid.

Ages at a stokvel meeting:

18

;

21

;

25

;

40

;

41

;

44

. Make a stem-and-leaf display and give the range.

Worked answer

The ages run from the teens to the forties, so use stems $1$ to $4$ .
Stem $1$ has leaf $8$ . Stem $2$ has leaves $1$ and $5$ .
Stem $3$ stays with no leaves. Stem $4$ has leaves $0$ , $1$ and $4$ .
Range: $44 - 18 = 26$ .

Answer: $26$

Every age is still there, so the extremes read off the two ends.

Traps

Finding the median before putting the data in order. Always order the data from smallest to largest first, then find the middle value.
Giving the smallest and largest values as the range. The range is one number: the largest value minus the smallest value.
Giving the frequency of the commonest value instead of the value itself. In $10$ ; $15$ ; $15$ ; $20$ ; $30$ the mode is $15$ , not $2$ .
Leaving out a stem that has no values, so the gap in the data disappears. Write every stem in turn, even one with no leaves beside it.

Learn and practise “Summarise a data set” in the app →

Grade 8 Collect, organize and summarize data

What gets asked

You must be able to

Traps that cost marks

Worked example

Collect and organise data

Examples

Traps

Summarise a data set

Rules to remember

Examples

Traps

About this material