Solve problems by analysing secondary one-variable data

Ontario Grade 11 Mathematics

Data Management

Analysing secondary one-variable data

A community report, a company spreadsheet, or a published survey may already contain useful data. When you analyse data gathered by someone else, you are analysing secondary data. This lesson focuses on data about one characteristic, such as delivery times or monthly rainfall. You will practise choosing a useful summary, checking what the values mean, and using the results to answer a question in context.

What you will learn

Explain what secondary data and one-variable data are.
Organize and describe a data set using its context and suitable summary measures.
Use information from a data set to answer a practical question without claiming more than the data show.

1. Start with the question and the source

A variable is a characteristic that can take different values. For example, the number of minutes a bus is late is a variable. A one-variable data set records values for just one characteristic. It might list late times for several buses, without recording another characteristic such as weather.

Secondary data are data that someone else has already gathered or published. Examples include a public report, a company’s past records, or a table in a news article. Before calculating anything, identify who gathered the data, what was measured, and which people, places, or time period the data cover. Those details help you decide whether the data can answer your question.

A useful analysis begins with a specific question. For example, a school might ask, “What was a typical wait time at the clinic last week?” The data need to be wait times from that clinic and period. If the source covers a different place or year, it may not answer the question well.

Secondary data can be useful, but the source and its limits matter. A published average alone does not show every recorded value. If you need to find a median or range, you need enough of the original data to calculate it. Do not treat missing information as if it were recorded.

Secondary data were gathered by someone else.
One-variable data describe one characteristic.
Check the source, measurement, group, and time period before drawing a conclusion.

2. Organize and summarize the values

First, check the units and make sure the values describe the same kind of measurement. Then sort the values from least to greatest. Sorting helps you locate the middle value and spot values that are unusually high or low.

The mean is the total of the values divided by the number of values. It uses every value, so one very large or small value can affect it. The median is the middle value after sorting. With an even number of values, it is the mean of the two middle values. The median can be a useful description when a value is far from most of the others.

The range is the greatest value minus the least value. It describes the full spread from one end of the data to the other, but it depends only on those two endpoints. The mode is the value that occurs most often. A data set can have no mode or more than one mode. These summaries describe different features, so do not assume one is always best.

Consider the ordered values

12

,

15

,

15

,

18

, and

25

minutes. The median is

15

minutes, the range is

13

minutes, and the mode is

15

minutes. The mean is

17

minutes. The mean is higher than the median because the value

25

pulls the total upward.

For a long list, a spreadsheet or calculator can help. You still need to check the variable, units, and result. A computer can calculate a mean for the wrong column just as quickly as for the right one. The mean is the sum of the values divided by their count. The range is the greatest value minus the least value.

Sort the data before finding the median.
Mean, median, mode, and range answer different questions.
Keep units with measures and conclusions.

3. Interpret results in context

A calculation becomes useful when you connect it back to the question. If a median wait is

15

minutes, say what waits were measured and where they came from. The number alone does not tell a reader whether the service was acceptable.

Look at the values as well as the summary. A mean can be pulled upward or downward by an unusual value. A median may better describe the centre in that situation, but it does not describe the full spread. The range can show how far apart the smallest and largest values are, but it does not show how many values lie between them.

Be careful with claims. Data about one clinic last week do not automatically describe every clinic or every week. Secondary data may be incomplete, old, or gathered for a different purpose. State what the data support, and avoid conclusions that go beyond the source.

When comparing two data sets, use the same kind of measure and check that the variables and units match. A smaller median may indicate a lower typical value, while a smaller range may indicate a narrower spread. Neither comparison alone explains why the values differ.

Choose a measure that fits the question and the data.
Use the source details and units in your written conclusion.
Describe only what the available data support.

Which measure helps answer which question?

Measure	What it describes	Useful question
Mean	Equal-share average of all values	What is the average recorded value?
Median	Middle value, or midpoint of the two middle values	What is the centre when a high or low value may affect the mean?
Mode	Most frequent value	Which value occurs most often?
Range	Difference between greatest and least values	How far apart are the endpoints?

Worked example

Comparing daily delivery times

A local delivery service publishes the times, in minutes, for six deliveries on one route:

18

,

22

,

19

,

45

,

21

, and

20

. Find the mean, median, and range. Which measure gives a more typical time, and what is one limit of this conclusion?

Sort the values
Put the times in order so the middle values and the endpoints are clear.
$18,\ 19,\ 20,\ 21,\ 22,\ 45$
Calculate the mean
Add all six delivery times and divide by six because the mean shares the total equally across the recorded deliveries.
$\frac{18+19+20+21+22+45}{6}=\frac{145}{6}\approx24.2$
Find the median
There are six values, so the median is halfway between the third and fourth values in the ordered list.
$\frac{20+21}{2}=20.5$
Find the range and interpret
Subtract the smallest time from the largest. Since $45$ is well above the other times, the median is a more typical centre for this list than the mean. These records cover only six deliveries on one route, so they do not establish typical times for every route or day.
$45-18=27$

Answer: The mean is about

24.2

minutes, the median is

20.5

minutes, and the range is

27

minutes. The median better represents a typical recorded time because the

45

-minute delivery raises the mean. The data describe only the six published deliveries.

Check: The ordered list has middle values

20

and

21

, and its endpoints are

18

and

45

. The median and range calculations match those values.

Worked example

Using published monthly rainfall

A regional weather page lists monthly rainfall totals, in millimetres, for six months as

34

,

51

,

47

,

62

,

28

, and

50

. A garden centre wants a simple description of the rainfall in those months. Find the mean, median, and range, then state what the figures do and do not tell the centre.

Order the rainfall totals
Sort the monthly totals before finding the middle and the spread.
$28,\ 34,\ 47,\ 50,\ 51,\ 62$
Find the mean
The six totals add to $272$ millimetres. Divide by six to find the average monthly total for these recorded months.
$\frac{34+51+47+62+28+50}{6}=\frac{272}{6}\approx45.3$
Find the median and range
The two middle totals are $47$ and $50$ , so their average is the median. The range is the largest total minus the smallest total.
$\frac{47+50}{2}=48.5,\quad 62-28=34$
Answer the practical question
The measures describe these six months only. They do not show when rain fell within each month or guarantee the same totals in another year. The centre should not treat the mean as a prediction for a future month.

Answer: For the six months, mean rainfall was about

45.3

millimetres, median rainfall was

48.5

millimetres, and the range was

34

millimetres. These summarize the published monthly totals, but they do not predict rainfall in a future month.

Check: The total is

272

millimetres. The ordered middle values are

47

and

50

, and the endpoints are

28

and

62

.

Common mistakes and how to avoid them

Finding the median before ordering the values.

Correction: Sort the list first. Then identify the middle value or the two middle values.

Calling the mean the typical value without checking the data.

Correction: Look for values far from the others. Compare the mean with the median before deciding which better describes the centre.

Treating a published data set as if it represents every place or time.

Correction: Name the group and period covered by the source. Keep conclusions within that scope.

Reporting a measure without units or context.

Correction: Include the measured variable and its units, such as “median wait time was

15

minutes.”

Lesson summary

Secondary data are existing data gathered by someone else.
One-variable data record one characteristic, such as time or rainfall.
Check the source and context, then organize the values before calculating.
Use mean, median, mode, or range according to the question.
Interpret results with units and do not claim more than the data show.

Check your understanding

Question 1

A published list of five repair times is

10

,

12

,

12

,

14

, and

27

minutes. What is the median?

$12$ minutes
$14$ minutes
$15$ minutes
$27$ minutes

Show answer and explanation

12

minutes

The values are already ordered. The third value of five is the middle value, so the median is

12

minutes.

Question 2

For the same repair times, what is the range?

$12$ minutes
$15$ minutes
$17$ minutes
$27$ minutes

Show answer and explanation

17

minutes

Subtract the least time from the greatest:

27-10=17

minutes.

Question 3

A report contains monthly electricity use for one home during a single year. Which conclusion is supported by the source?

The values describe that home's recorded monthly use for that year.
The values prove every home uses the same amount of electricity.
The values predict the exact use of that home next year.
The values describe electricity use in every home in the region.

Show answer and explanation

The values describe that home's recorded monthly use for that year.

The data cover one home and one year. They do not establish what every home uses or guarantee a future amount.

Key terms

Variable: A characteristic that can have different values, such as time or rainfall.
One-variable data: Values recorded for one characteristic.
Secondary data: Data that someone else has already gathered or published.
Median: The middle value of an ordered list, or the mean of the two middle values when the list has an even number of values.
Range: The greatest value minus the least value.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation D1.10. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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