In this unit students investigate the different ways data varies over time looking at variables that fluctuate randomly, those that steadily increase or decrease and those that show seasonable behaviour. They display data, use appropriate vocabulary and determine appropriate statistics.

- plan an investigation
- be able to display time series data
- discuss features of time series distributions
- compare features of time series distributions
- report on possible sources of error and limitations in displayed data.
- report the results of a statistical investigation concisely and coherently

This unit involves students analysing both given and collected statistics in a variety of ways. The terminology associated with different time series is explored. Students choose a time series and, working in groups, formulate questions about the data they have collected and report back their findings.

Time series are investigated for both discrete and continuous data.

A graphics calculator or spreadsheet would be useful here to help analyse the data.

Four sets of time series data (see Sessions 1, 2, 3)

Answers to questions.

time series data, continuous data, discrete data, grouped data, frequency polygon, distribution, line graph, base data, index, suppression of zero, stretching scales, guarantee

#### Sessions 1, 2, 3

Here we explore the four sets of time series data listed in the resources above.

Note: It is anticipated that up to three sessions might be spent on analysing the four data sets below.

#### 1. Traffic Department Statistics

Present students with the Traffic Department statistics concerning the length of time people stay in hospital after accidents.

Length of stay (days) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10-19 | 20-29 | 30+ |

% of those admitted | 35 | 20 | 5 | 6 | 2 | 1 | 3 | 1 | 1 | 9 | 6 | 11 |

#### NZ Traffic Department

Check students' understanding of the table with questions like

*What percentage of people stay in hospital five days?**What percentage of people stay in hospital less than three days? More than three days?**Why do the percentages first drop, then rise?**Why can we only estimate the percentage of people who stay in hospital 12 days? What is a reasonable estimate of the percentage of people who stay in hospital 12 days?**Why is some of the data grouped (such as those who stay in hospital 10-19 days)?**To what level has the data been rounded?**Do people ever stay 3.5 days? What does this mean?**What does the table tell us in terms of the number of hours people stay in hospital after accidents?**What is the mode, what does it tell us?**What is the overall shape of the distribution given by the number of days people stay in hospital after accidents?**If you were to display the data in the form of a scatter graph or frequency (percentage) polygon how would you overcome the problem of the grouped data towards the right of the distribution?*Although calculating the mean of data of this type is not required, a discussion of the irrelevance of using the mean might be attempted.

Students display the first nine days of the data in the form of both a scatter graph and a line graph with the percentages of people admitted to hospital on the y-axis and length of stay on the x-axis. They may draw them by hand, use a graphics calculator or use a spreadsheet. Students suggest ways in which the data may be displayed for hospital stays beyond that time.

Discuss reasons for and against joining the points on the graph, the concept of showing a trend without giving meaning to in-between points.

This data is discrete since it is based on the number of days people stay in hospital and yet time is a continuous variable. Discuss.

#### Health Department Statistics

Present students with Health Department statistics of median heights for girls aged 0 to 16 years

Age (years)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Height (cm)

50

67

81

95

103

107

113

119

125

130

136

144

150

155

159

161

162

__Department of Health__Check their understanding of the table with questions like:

*What does ‘median heights’ mean here?**What proportion of girls are 119cm or more at age 7?**At what age are 50% of girls under 130cm?**The heights have been rounded. How?*Further discuss the differences between discrete and continuous data using the two data sets already presented as examples.

Give more examples of data which varies continuously.

*The table is given for the median heights of*girls*at various ages. Why?*Students plot a line graph of the data in the table and describe its shape. They may draw them by hand, use a graphics calculator or use a spreadsheet.

Discuss reasons for and against joining the points on the graph. Meaning can be given to values between those plotted. Discuss.

*How would the graph continue to the right for the heights of older girls?*

#### 3. Reserve Bank Data

Present students with the Reserve Bank data for the Trade-Weighted Index for the New Zealand dollar.

Day

1

2

3

4

5

6

7

8

9

10

TWI

67.7

67.8

67.3

66.1

66.2

66.4

66.4

66.9

66.4

66.0

__Reserve Bank of New Zealand__The Trade-Weighted Index (TWI) is a measure of the New Zealand dollar relative to the currencies of NZ's major trading partners.

Check students' understanding with questions like:

*What is an index? What is meant by ‘base data’?**How would you describe the fluctuations in the TWI over the 10 days?**Given that the fluctuations are small, what difficulties does this imply for displaying the data?**How might the difficulties be overcome?**Is the data discrete or continuous?*Discuss the effect on data display of the various ways scales may be altered to accommodate relatively small fluctuations. Discuss the accepted methods for these (suppression of zero, stretching scales) and the misinformation this might imply. A homework exercise might be for students to collect poorly displayed data from newspapers and magazines - there's a wealth to choose from.

*Does the data show a meaningful trend over the 10 days?**Given that the base rate was 100 in April 1991, what does this tell us about the way the NZ dollar has performed since then?**What comments can we make about the future shape of the distribution?*Students display the data in the table in the form of both a scatter graph and a line graph and describe its shape. They may draw them by hand, use a graphics calculator or use a spreadsheet.

Discuss reasons for and against joining the points on the graph, the concept of showing short term trends without giving meaning to in-between points.

#### 4. Movietime Data

Present students with data for total ticket sales for the 21 days, Sunday, June 6 to Saturday, June 26, 2004 for the cinema complex Movietime 4.

Date(June)

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

Tickets sold

622

205

271

583

710

1416

1784

812

407

590

608

802

1196

1345

310

414

415

834

1221

1468

1582

Check students' understanding with questions like:

*What does ‘total daily ticket sales’ mean here?*

How many movie tickets were sold on June 13^{th}?

Over what days are ticket sales increasing?

Do you think more or less tickets were sold on 27^{th}June than 26^{th}?

Could it be guaranteed that fewer tickets would be sold on Monday 26th than Sunday 27th?

Could it be guaranteed that fewer tickets would be sold on any Monday than would be sold on the Saturday before?Students plot a line graph of the data and discuss its main features. They may draw them by hand, use a graphics calculator or use a spreadsheet.

There seems to be a pattern in the data points.

*What is the pattern and why does it occur?*

Would this pattern persist throughout the whole year?List some occasions when the pattern might break down.

Students brainstorm and put together a list of time series data that show cycles or seasonal variations

*.*

#### Session 4

In this session we discuss the concept of time series data in general.

Discuss the nature of time series and brainstorm for other examples.

Categorise time series into discrete and continuous types and discuss their differences.

Using student's suggestions for time series, discuss ways in which data may be collected.

*For continuous data, discuss what factors govern the regularity of taking measurements? For example: if we are investigating the height of a bean plant over time, should we measure every hour, day, week, month?*

For discrete data, discuss what factors govern the unit of time used to count items. For example: if we are looking at school absences, should we record them every half day, day, week, term?Posing questions: encourage students to think of questions they would like answered about some of the time series they have suggested. For example:

*Are there more absences from school on Fridays? Mondays?**Do bean plants grow at a steady rate?**Are people more like to die in the winter?*Students, working in small groups, choose a time series they would like to study. They design a method of collecting appropriate data, decide on the questions about the time series they would like answered and ways they can present their findings in a report. They list the tasks required and allocate them to members of the group. There are two options here. Students could be asked to collect time series data themselves, in which case you will need to provide a sufficient gap between this session and Session 5 so that they will have the opportunity to collect the data (a week should be long enough). Alternatively, students could be asked to find time series data that has been collected in other sources, such as the internet, or the newspaper.

#### Session 5

In this session the students complete the work of Session 4, report back and consider some open questions.

Students work on their own collected data from Session 4 and report back their group findings (orally or in poster form).

Despite our analyses of time series data there are a number of questions left unanswered.

*How do we know whether any patterns that have emerged will continue?**How can we be sure that over longer time periods other patterns might not arise?*Consider any other questions that occur to the class.