Choosing a good sample, and calculating the mean, median, mode and range of data.
Practise Collect, organize and summarize data in the app →
What gets asked
- Choose and justify an appropriate sample for a survey.
- Calculate the mean, median or mode of a data set.
- Calculate the range of a data set.
- Identify an extreme value or an outlier in a data set.
You must be able to
- Explain why a sample must represent the whole population fairly.
- Order data from smallest to largest before finding the median.
- Average the two middle values when a data set has an even number of values.
- Subtract the smallest value from the largest to find the range.
Traps that cost marks
- Finding the median without first putting the data in order. Always order the data from smallest to largest before finding the middle value.
- Picking the middle value of an even-numbered data set instead of averaging the two middle values. With an even number of values, add the two middle ones and divide by 2.
- Giving the two extreme values as the range instead of their difference. The range is one number: the biggest value minus the smallest value.
Worked example
- Order the values: 9; 12; 15; 18; 21.
- With 5 values, the median is the middle, 3rd, one.
- The median is 15.
Sources and samples
Who the question is really about, who you actually ask, and how the asking can spoil the answer.
- population
- Every single person or thing that the question is about.
- sample
- A smaller group, chosen out of the population to stand for all of it.
- representative
- A sample is representative when it looks like the population in the ways that matter.
- bias
- Anything in the way data is gathered that pushes the answers one way.
- questionnaire
- A set of written questions handed to people to fill in.
- leading question
- A question worded so that it pushes the reader towards one answer.
Start by deciding who or what the question is about. That whole group is the population. If the question asks about learners at your school, the population is every learner at your school.
You can seldom ask everyone. You ask a sample and let it speak for the population. A sample is only worth having when it is representative.
Match the source to the question. A survey in your own class cannot tell you what the whole country thinks. A national figure needs a national source, such as Statistics South Africa.
Asking whoever is easiest to reach is not the same as choosing fairly. Ask only your own friends about music and you leave out everybody whose taste differs from theirs.
The method matters as much as the sample. A questionnaire asking 'Do you agree the tuck shop is too dear?' pushes people to say yes. That wording is bias, and it spoils the answers before you start.
Examples
Worked answer
- The population here is every shop in the country.
- Five shops in one street cannot speak for all of them.
- Prices differ from province to province and from town to town.
- Use a national source, such as Statistics South Africa.
Answer: no, use a national source
The source has to reach as widely as the question does.
Worked answer
- The population is all learners at the school.
- The sample is learners who already chose soccer.
- So the sample is not representative of the school.
- Better: pick learners at random off the class lists.
Answer: the sample is not representative
The group chosen had already agreed with one answer before anyone asked.
Worked answer
- The wording pushes the reader towards yes.
- That is a leading question, and it creates bias.
- Take out the pushing words and leave both sides open.
- Ask instead: 'How long should the school day be?'
Answer: How long should the school day be?
A fair question never tells you which answer to give.
Traps
- Using a class survey to answer a question about the whole country. Match the source to the question. A national figure needs a national source.
- Treating the easiest group to reach as if it stood for everybody. Ask how the group was chosen. Convenient is not the same as representative.
- Calling a group a sample when in fact every member of it was asked. If you asked everyone, that group is the population, not a sample.
- Wording a question so that it pushes people towards one answer. Take out the pushing words. A leading question gives you bias, not data.
Averages and spread
Three ways to give the middle of a set of numbers, and one way to say how spread out they are.
- 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 put in order from smallest to largest.
- mode
- The value that appears most often. A set can have more than one, or none.
- range
- The largest value minus the smallest. It is one number, not two.
- outlier
- A value far away from the rest. It is not simply the biggest one.
The mean uses every value, so one very large number pulls it up. The median only cares about position, so it is not pulled about in the same way.
Put the numbers in order before you look for the median. An unordered list gives the wrong middle almost every time.
If there is an even count, there are two middle values. The median is the mean of those two, so it may be a number that is not in the list at all.
The range is a single number: largest minus smallest. A value sitting far away from all the others is an outlier, and it stretches the range without moving the median much.
Sorting by two things at once needs a two-way table. Every item is counted once, in exactly one cell, so the row and column totals must both add up to the number of items.
Rules to remember
- mean
Examples
Worked answer
- Mean: , and .
- Order them: ; ; ; ; .
- There are five values, so the middle one is the third: .
- appears twice and nothing else repeats, so the mode is .
Answer: mean R21, median R18, mode R18
The mean is higher than the median because one fare of R30 pulls it up.
Worked answer
- Order them: ; ; ; ; ; .
- There are six values, so two sit in the middle.
- They are and , so the median is .
- .
- Range: .
Answer: median 14,5 and range 9
With an even count the median can be a value nobody actually scored.
Worked answer
- Grade 9 total: .
- So the Grade 8 total is .
- Of those, walk.
- So Grade 8s do not walk.
Answer: 4 learners
Every learner sits in exactly one cell, so the four cells add to 30.
Traps
- Finding the median without putting the numbers in order first. Order the list from smallest to largest, then count in to the middle.
- Giving the smallest and largest values as the range. The range is one number, the difference: .
- Calling the largest value in a set an outlier just because it is the largest. An outlier sits far from all the rest. In 17; 18; 18; 22; 30 nothing is far enough away.
- Counting a learner under both criteria, so the table totals come to more than the class. Each item belongs in exactly one cell. Check that the cells add up to the total.