Chapter 7 of 8

All 8 chapters
  1. 1Samples and bias, readunderstood
  2. 2Averages from frequency tables, readunderstood
  3. 3Cumulative frequency, readunderstood
  4. 4Box plots, readunderstood
  5. 5Histograms with unequal widths, readunderstood
  6. 6Scatter graphs and correlation, readunderstood
  7. 7Data over time, readunderstood
  8. 8Telling the truth with data, readunderstood

M4.7.7 Chapter 7

Data over time

The rain in a town rises and falls every year. How can I see a slow change over the years, and the ups and downs that repeat?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Plot a time series and describe the trend
  • Spot ups and downs that repeat in a time series

A time series is a set of readings taken at regular times, such as each day or each quarter of a year. The order matters. A town records its rain in millimetres for each of 12 quarters, three years in all.

The rain in a town in each of 12 quarters. The head is quarter, rain (mm). Row 1: 1, 80. Row 2: 2, 40. Row 3: 3, 20. Row 4: 4, 60. Row 5: 5, 86. Row 6: 6, 44. Row 7: 7, 24. Row 8: 8, 66. Row 9: 9, 94. Row 10: 10, 48. Row 11: 11, 26. Row 12: 12, 72.quarterrain (mm)180240320460586644724866994104811261272
Quarters 1 to 4 are year 1, 5 to 8 are year 2, and 9 to 12 are year 3.

Draw it first

Plot the time series

  1. The x axis is quarter from 0 to 12 and the y axis is rain (mm) from 0 to 100. The 12 dots are at heights 80, 40, 20, 60, 86, 44, 24, 66, 94, 48, 26 and 72 for quarters 1 to 12. Step 1 shows the dots. Step 2 joins them with straight lines. Step 3 adds a dashed straight line that rises slowly through the middle of the readings, and three dots on it named 50, 55, 60. They are the yearly means, at quarters 2.5, 6.5 and 10.5.24681012204060801000quarterrain (mm)

    1 Put time across and the reading up. Plot one dot for each quarter.

  2. The x axis is quarter from 0 to 12 and the y axis is rain (mm) from 0 to 100. The 12 dots are at heights 80, 40, 20, 60, 86, 44, 24, 66, 94, 48, 26 and 72 for quarters 1 to 12. Step 1 shows the dots. Step 2 joins them with straight lines. Step 3 adds a dashed straight line that rises slowly through the middle of the readings, and three dots on it named 50, 55, 60. They are the yearly means, at quarters 2.5, 6.5 and 10.5.24681012204060801000quarterrain (mm)

    2 Join the dots in order with straight lines. The joining lines show the order, not new readings.

  3. The x axis is quarter from 0 to 12 and the y axis is rain (mm) from 0 to 100. The 12 dots are at heights 80, 40, 20, 60, 86, 44, 24, 66, 94, 48, 26 and 72 for quarters 1 to 12. Step 1 shows the dots. Step 2 joins them with straight lines. Step 3 adds a dashed straight line that rises slowly through the middle of the readings, and three dots on it named 50, 55, 60. They are the yearly means, at quarters 2.5, 6.5 and 10.5.50556024681012204060801000quarterrain (mm)

    3 Draw a dashed line through the middle of the readings, to show the general direction.

Two things stand out. First, the readings go up and down every year: high at the start, low in the third quarter. Second, each year as a whole is a little wetter than the one before.

The trend is the general direction of the readings over a long time. Here it is a slow rise. A seasonal pattern is a rise and fall that repeats after the same number of readings. Here it repeats every 4 quarters.

Worked example

Find the mean rain in each of the three years.

  1. Year 1: 80 + 40 + 20 + 60 = 200 mm. The mean is 200 ÷ 4 = 50 mm.
  2. Year 2: 86 + 44 + 24 + 66 = 220 mm. The mean is 220 ÷ 4 = 55 mm.
  3. Year 3: 94 + 48 + 26 + 72 = 240 mm. The mean is 240 ÷ 4 = 60 mm.

Answer: The yearly means are 50 mm, 55 mm and 60 mm

A bar chart with the y axis rain (mm) and three bars: Year 1 with a height of 50, Year 2 with 55 and Year 3 with 60.0102030405060Year 1Year 2Year 3rain (mm)
The yearly means rise by 5 mm each year.

A mean over a whole year averages out the seasonal ups and downs. What is left is the trend. The yearly means are 50, 55 and 60 mm. They are the dots on the dashed line in the first graph, at the middle of each year.

The rain in each year, one row for each year, shown as yr, and one column for each quarter. The columns are quarters 1 to 4. The head is yr, 1, 2, 3, 4. Row 1: 1, 80, 40, 20, 60. Row 2: 2, 86, 44, 24, 66. Row 3: 3, 94, 48, 26, 72.yr1234180402060286442466394482672
Read down a column. Quarter 1 is always the wettest. Quarter 3 is always the driest.

Worked example

Which quarter has the lowest rain in every year? How big is the gap between the highest and the lowest quarter in year 1?

  1. Year 1 readings are 80, 40, 20 and 60. The lowest, 20, is in quarter 3.
  2. Year 2 readings are 86, 44, 24 and 66, and year 3 readings are 94, 48, 26 and 72. Again the lowest is in quarter 3.
  3. So the pattern repeats: quarter 1 is the wettest and quarter 3 is the driest each year.
  4. In year 1 the gap is 80 − 20 = 60 mm.

Answer: Quarter 3, and the gap is 60 mm

We can use the trend to look a little way ahead. The means rise by 5 mm each year. If that carries on, year 4 has a mean near 65 mm. This is a step beyond the data, so it is a cautious estimate. The further we go, the less we should trust it.

Worked example

Use the yearly means 50, 55 and 60 mm to estimate the mean of year 4.

  1. The first rise is 55 − 50 = 5 mm.
  2. The second rise is 60 − 55 = 5 mm too. The trend is a rise of 5 mm a year.
  3. If the trend carries on, year 4 has a mean of 60 + 5 = 65 mm.
  4. Year 4 is only one year beyond the data, so this is a cautious estimate, not a promise.

Answer: About 65 mm

See it three ways

With things

A seedling is measured every week for 8 weeks. Its heights in centimetres are 4, 9, 14, 18, 24, 29, 33 and 39.

As a picture

A time series with the x axis week from 0 to 8 and the y axis height (cm) from 0 to 40. Eight dots at weeks 1 to 8 rise steadily: (1, 4), (2, 9), (3, 14), (4, 18), (5, 24), (6, 29), (7, 33) and (8, 39). A dashed straight line runs through them.12345678102030400weekheight (cm)

In words

The trend is a steady rise of about 5 cm each week. There is no seasonal pattern: nothing goes up and down in a repeating way.

In symbols

The dashed line is y = 5x − 1. Each week adds about 5 cm.

Try it

Count something each day for 14 days

You need: paper and a pencil, something you can count each day, such as glasses of water.

  1. Each evening, write the day number and the count. Do this for 14 days.
  2. Draw a graph with the day across and the count up. Plot a dot for each day and join them.
  3. Look for a trend. Then look for a pattern that repeats, such as a high count at the end of each week.

What you will see: A pattern that repeats every 7 days is common, because many daily habits follow the week. Here, days 6 and 13 are the highest.

If you cannot try it, this is what happens.
A time series with the x axis day from 0 to 14 and the y axis glasses from 0 to 10. Fourteen dots are joined by lines: 6, 5, 6, 6, 7, 9, 8 for days 1 to 7 and 6, 6, 5, 7, 7, 10, 8 for days 8 to 14. The highest dots are on days 6 and 13.24681012142468100dayglasses

A wrong idea many people have

If the readings go up and down, there is no trend.

Why it feels right: A line that zigzags does not look like it is going anywhere.

The test that breaks it: The rain readings zigzag every year. But the yearly means are 50, 55 and 60. Each year is wetter than the last.

What is true: A trend is the general direction. Ups and downs that repeat sit on top of the trend. Averaging over a whole cycle shows the trend.

Key words

time series
readings taken at regular times, plotted in order.
trend
the general direction of a time series over a long time.
seasonal pattern
a rise and fall that repeats after the same number of readings.

Back to your guess

The answer is C: Plot each reading against time, join the points, and look for the general direction and for ups and downs that repeat.

A mean of all the readings hides the order in which they happened. Plotting each reading against time keeps the order. Then we can see the general direction, called the trend, and the ups and downs that repeat, called a seasonal pattern.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The graph shows the number of visitors to a park each week. Which describes the trend?

    A time series with the x axis week from 0 to 8 and the y axis visitors from 0 to 100. Eight dots fall steadily from 90 in week 1 to 55 in week 8: 90, 84, 80, 75, 70, 66, 60 and 55.12345678204060801000weekvisitors
    Tap your answer.

    1. Not this one. The dots start high and end lower. A rising trend would climb from left to right. Go to M4.7.7. Try another.

    2. Not this one. Numbers always change a little. The trend is the general direction, and here the dots fall steadily. Go to M4.7.7. Try another.

    3. Yes, that is right. The visitors drop from 90 in week 1 to 55 in week 8. The general direction is down.

    See the answer

    The trend is a fall.

    The visitors drop from 90 in week 1 to 55 in week 8. The general direction is down.

    • If you chose A: The dots start high and end lower. A rising trend would climb from left to right. Go to M4.7.7.
    • If you chose B: Numbers always change a little. The trend is the general direction, and here the dots fall steadily. Go to M4.7.7.

  2. This graph shows the visitors to a library on each of 14 days. After how many days does the pattern repeat?

    A time series with the x axis day from 0 to 14 and the y axis visitors from 0 to 100. Fourteen dots are joined by lines: 40, 45, 42, 48, 60, 90, 20 for days 1 to 7 and 42, 46, 44, 50, 62, 94, 22 for days 8 to 14. The highest dots are on days 6 and 13.2468101214204060801000dayvisitors
    Tap your answer.

    1. Not this one. 6 is the day of the first peak. The question asks how many days pass before the peak comes again. Go to M4.7.7. Try another.

    2. Yes, that is right. The peaks are on days 6 and 13, which are 7 days apart. The low points on days 7 and 14 are also 7 days apart.

    3. Not this one. 14 is the number of days in the whole series. The pattern shows twice in it, so it repeats after half of that. Go to M4.7.7. Try another.

    4. Not this one. The readings do change from day to day, but the repeating shape is the whole rise and fall, which takes longer. Go to M4.7.7. Try another.

    See the answer

    7 days

    The peaks are on days 6 and 13, which are 7 days apart. The low points on days 7 and 14 are also 7 days apart.

    • If you chose A: 6 is the day of the first peak. The question asks how many days pass before the peak comes again. Go to M4.7.7.
    • If you chose C: 14 is the number of days in the whole series. The pattern shows twice in it, so it repeats after half of that. Go to M4.7.7.
    • If you chose D: The readings do change from day to day, but the repeating shape is the whole rise and fall, which takes longer. Go to M4.7.7.

  3. Which time series is most likely to show a pattern that repeats?

    Tap your answer.

    1. Yes, that is right. Pool visits follow the seasons. The same months are busy each year, so the pattern repeats every 12 months.

    2. Not this one. A growing tree has an upward trend, but its height does not go up and down in a repeating way. Go to M4.7.7. Try another.

    3. Not this one. The ball keeps rolling further, then slows. It does not rise and fall in a repeating way. Go to M4.7.7. Try another.

    See the answer

    The number of visitors to an outdoor swimming pool each month for several years.

    Pool visits follow the seasons. The same months are busy each year, so the pattern repeats every 12 months.

    • If you chose B: A growing tree has an upward trend, but its height does not go up and down in a repeating way. Go to M4.7.7.
    • If you chose C: The ball keeps rolling further, then slows. It does not rise and fall in a repeating way. Go to M4.7.7.

  4. The rain in the four quarters of a year is 88, 52, 28, 72 mm. What is the mean rain for the year?

    Tap your answer.

    1. Not this one. 240 is the total of the four readings. The mean is the total shared out over the 4 quarters. Go to M4.7.7. Try another.

    2. Not this one. 62 is the median, halfway between the two middle readings. The mean uses all four readings added up. Go to M4.7.7. Try another.

    3. Yes, that is right. 88 + 52 + 28 + 72 = 240. Then 240 ÷ 4 = 60.

    See the answer

    60 mm

    88 + 52 + 28 + 72 = 240. Then 240 ÷ 4 = 60.

    • If you chose A: 240 is the total of the four readings. The mean is the total shared out over the 4 quarters. Go to M4.7.7.
    • If you chose B: 62 is the median, halfway between the two middle readings. The mean uses all four readings added up. Go to M4.7.7.

  5. The yearly means of a time series are 40, 44, 48. If the trend carries on, what is the estimate for the next year?

    Tap your answer.

    1. Not this one. 4 is how much the mean rises each year. The next mean is that rise added to the last mean. Go to M4.7.7. Try another.

    2. Not this one. You added the rise twice. The next year is only one step on: 48 + 4. Go to M4.7.7. Try another.

    3. Not this one. That is the last mean. The trend says the mean keeps rising, so the next one is higher. Go to M4.7.7. Try another.

    4. Yes, that is right. The means rise by 4 each year. The next is 48 + 4 = 52.

    See the answer

    52

    The means rise by 4 each year. The next is 48 + 4 = 52.

    • If you chose A: 4 is how much the mean rises each year. The next mean is that rise added to the last mean. Go to M4.7.7.
    • If you chose B: You added the rise twice. The next year is only one step on: 48 + 4. Go to M4.7.7.
    • If you chose C: That is the last mean. The trend says the mean keeps rising, so the next one is higher. Go to M4.7.7.

  6. What does the trend of a time series show?

    Tap your answer.

    1. Not this one. The highest reading is one value. The trend is about the whole series. Go to M4.7.7. Try another.

    2. Not this one. That is the seasonal pattern. The trend is the general direction under those ups and downs. Go to M4.7.7. Try another.

    3. Yes, that is right. The trend says whether the readings are mostly rising, falling or staying level.

    See the answer

    The general direction of the readings over a long time.

    The trend says whether the readings are mostly rising, falling or staying level.

    • If you chose A: The highest reading is one value. The trend is about the whole series. Go to M4.7.7.
    • If you chose B: That is the seasonal pattern. The trend is the general direction under those ups and downs. Go to M4.7.7.

  7. From an earlier chapter

    A line of best fit has the rule y = 3x + 2. Estimate y when x = 9.

    See the answer

    29

    y = 3 × 9 + 2 = 27 + 2 = 29.

  8. The four quarterly readings of one year are 12, 30, 22 and 8. What is the mean for the year?

    See the answer

    18

    12 + 30 + 22 + 8 = 72. Then 72 ÷ 4 = 18.

Keep it

Copy this into your exercise book.

Time series: readings in order, time across. Trend: the general direction. Seasonal pattern: a rise and fall that repeats. The mean over a whole cycle shows the trend.

Draw: A zigzag line that rises slowly, with a dashed line through the middle and two matching peaks marked.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.1 MB). It opens on any phone.