Seasonal Trends, Rates of Change and Index Numbers

Numbers that describe change

You have already learned to smooth a wobbling series with moving averages, to draw a line of best fit, and to separate out the seasonal variation. This module is about what you do next: turning a series of numbers into a statement about change that somebody could act on. There are three of them here, and at first they look unrelated. Index numbers express a value as a comparison with an earlier one. Rates of change say how fast something is rising or falling. Predictions say what a trend suggests will happen next. Here is the idea that connects all three, and it is worth holding on to because it is where the marks in this section are. Each one is only as good as what it compares against. An index number of 115 means nothing at all until you know the base year it is measured from. A rate of change means nothing until you know the interval it was measured over. And a prediction means nothing unless the trend it rests on keeps going, which is an assumption rather than a fact. So the arithmetic in this module is straightforward, and it is not what is being tested. What is being tested is whether you know what your number is comparing to, and whether you say so.

Words for change over time

Six terms. The named indices are on your specification as examples, so recognise them; you are not expected to know their current values.

What an index number tells you

An index has a base year of 100 and its value this year is 115. What does that tell you?

• That the quantity is 15% higher than it was in the base year
• That the quantity has a value of 115
• That the quantity has risen by 115%
• Nothing, without knowing the units the quantity is measured in

Work out the index number

In the base year a quantity was 40. This year it is 46. Calculate the index number for this year, taking the base year as 100. Give the number only.

Everything is measured from the base

The base year is the part students leave out, and leaving it out is what costs marks. An index is a comparison, so it needs something to compare with. One year in the series is chosen as the base and given the value 100. Every other year is then expressed as a percentage of that year: index number = (current value divided by base year value) × 100. Three consequences follow, and each of them turns up in questions. First, the same data gives different index numbers if you choose a different base year. The underlying figures have not changed; only what they are being compared with has. Second, two indices with different base years cannot be compared directly. If one series is based on one year and another on a different year, the numbers are not on the same scale, and saying "this index is higher so this rose more" is simply wrong. Third, an index above 100 means the quantity is above its base-year level, below 100 means below it, and exactly 100 means unchanged from the base. Notice how much of this is about saying what you are comparing to. ⚠️ A note on tiers, because the wording matters: basic index numbers, including RPI, CPI and GDP as named examples, are Higher content that appears on BOTH tiers. Weighted index numbers are Higher only. Do not assume that because a topic is labelled Higher it will not appear on your paper.

Match each index value to its meaning

Which two are true of index numbers

Select the TWO accurate statements about index numbers.

• Changing which year is used as the base changes every index number in the series
• Two indices with different base years cannot be compared directly
• An index number tells you the actual size of the quantity being measured
• An index can never fall below 100

Basic, and weighted

Two kinds of index number, and your specification treats them differently by tier. Read the last row of each column carefully.

A weighted index, worked

A weighted index combines several indices into one, letting the more important items count for more. Here is one worked in full. The data: three categories of spending, each with its own index number and a weight showing how much of the household budget it accounts for. Housing has an index of 110 with a weight of 5. Transport has an index of 125 with a weight of 3. Clothing has an index of 95 with a weight of 2. The formula: weighted index = the sum of (each index × its weight), divided by the sum of the weights. Step one, multiply each index by its weight. Housing: 110 × 5 = 550. Transport: 125 × 3 = 375. Clothing: 95 × 2 = 190. Step two, add those products. 550 + 375 + 190 = 1115. Step three, add the weights. 5 + 3 + 2 = 10. Step four, divide. 1115 ÷ 10 = 111.5. Now the part that earns the interpretation marks. The overall index is 111.5, meaning spending across these categories is about 11.5% higher than in the base year. Notice that this sits closer to the housing figure than to the transport one, even though transport rose most. That is the weighting doing its job: housing carries the largest weight, so it pulls the overall figure towards itself. If you had simply averaged the three indices you would have got a different answer, and a misleading one, because it would have treated a small category as mattering as much as a large one. Saying why the weighting matters is worth as much as the arithmetic.

The order to weight it in

Put the steps of calculating a weighted index number into the correct order.

• Identify each item's index number and its weight
• Multiply each index number by its own weight
• Add up all of those products
• Add up all of the weights
• Divide the total of the products by the total of the weights
• Interpret the result as a comparison with the base year, saying which items pulled it

Rates of change, and what comes next

A rate of change says how fast something is rising or falling, and it always has a per-something in it: per year, per month, per day. There are two ways you will meet it. From a graph, which is Foundation content: the steeper the line, the faster the change. A line rising steeply means a rapid increase, a shallow line a slow one, a horizontal line no change at all, and a line falling means a decrease. You should be able to say which period on a graph shows the fastest change, and describe what that means in context, whether the graph shows births, deaths, house prices or unemployment. From a table, which is Higher content: you calculate it, and the formula will be given to you in the question. For a simple case it is the change in the quantity divided by the time it took. If a population rises from 6000 to 7200 over four years, the change is 1200 and the rate is 1200 ÷ 4 = 300 per year. If a price rises from 190 to 250 over three years, the rate is 60 ÷ 3 = 20 per year. Always state the units, because a rate without them is not an answer. Then there is prediction, which is Higher only. If a trend has been established, you can extend it to say what the next value is likely to be. But be careful how you say it, because the marks are in the caution: a prediction assumes the trend continues, and that assumption is exactly what a good answer states rather than hides. The further ahead you predict, the weaker the assumption becomes.

The formulae in words

An index number is calculated as the current value divided by the _____ year value, multiplied by _____. An index of exactly 100 means the quantity is _____ from the base year. A weighted index is the sum of each index multiplied by its _____, divided by the sum of the weights. A rate of change is the change in the quantity divided by the _____ taken, and must be given with its units.

The claim that forgets the base

Four statements about a series of index numbers. Select the ONE that is wrong.

• The index rose from 100 to 115, so the quantity is 15% higher than in the base year.
• Our index is 120 and theirs is 108, so ours has risen more since its base year than theirs has since its own.
• The index fell below 100, which means the quantity is now lower than it was in the base year.
• Changing the base year would change every index number in the series, even though the underlying data is the same.

Say what the prediction rests on

Assemble a prediction that states its own assumption.

Reading change in context

You are given a table and a graph about a town and asked to interpret them. Work through the decisions.

• A graph shows house prices rising, with one section much steeper than the rest. What does that section show?
• A table gives population figures. You calculate a rate of change and get the number 300. What must you add?
• Two towns quote price indices of 120 and 108, but with different base years. What can you conclude?
• You are asked to predict next year's figure from the trend. How should you word it?

Explain index numbers and rates of change

Explain what index numbers and rates of change are, how each is calculated, and what has to be stated alongside each one for it to mean anything.

• Explain what an index number is and give the formula, including the role of the base year
• Explain what an index of 100, of 110 and of 90 each tell you
• Explain why two indices with different base years cannot be compared directly
• Explain what a weighted index number adds, and give its formula
• Explain how a rate of change is read from a graph and calculated from a table
• Explain why a rate of change must always be given with its units
• Explain what a prediction from a trend assumes, and why confidence falls the further ahead it reaches
• Finish by explaining what all three have in common: each is only as good as what it compares against