Topics

Processing data: percentages, indexes and real values

Covers AS91226

What "processing" means

The Achieved criterion says "processing and presenting statistical data ... to show trends".
Processing means doing something to the raw numbers so a trend becomes visible: converting to percentages, building an index, adjusting for inflation or population, or calculating a change.
Copying a Stats NZ table into your report is not processing. You must transform it.

Percentage change

$\% \text{ change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100$

Converts levels into rates of change, which is what most economic trends are about.
Always say what period the percentage covers — over a quarter, or over a year.

Percentage change versus percentage point change

If unemployment goes from 4.0% to 5.0%, that is a rise of 1.0 percentage point, and a 25% increase in the rate.
Both are correct; they mean different things. Say which one you are using.

Index numbers

An index rescales a series so every value is compared to a base period, which is set to 100 or 1000.

$\text{index} = \frac{\text{value in year}}{\text{value in base year}} \times 1000$

Why indexes are useful here

They strip out the units, so two series measured in different things — dollars and people, say — can be plotted on the same graph and compared directly.
That is exactly what the inter-relationship section needs.

Real values

Converting a nominal series to a real one removes the effect of price changes:

$\text{real value} = \text{nominal value} \times \frac{\text{price index in base year}}{\text{price index in current year}}$

Use it for GDP, wages, incomes and government spending — any series measured in dollars over time.
Approximation for rates of change:

$\text{real change} \approx \text{nominal change} - \text{inflation rate}$

Per capita values

$\text{per capita} = \frac{\text{total}}{\text{population}}$

Essential when population is changing. Real GDP can rise 2% while population rises 3%, so real GDP per capita falls — the average person is worse off even though the economy grew.
Any comparison over a long period, or between regions of different sizes, should be per capita.

Averages and moving averages

A moving average smooths a volatile series so the underlying trend shows through.
A four-quarter moving average averages each quarter with the three before it, which removes most seasonal noise.
Useful when the raw series jumps around so much that the trend cannot be seen.

Which processing method for which job

You want to…	Use
Show how fast something is changing	Percentage change
Compare two series measured in different units	Index numbers
Remove the effect of inflation	Real values
Account for a changing population	Per capita
See through a volatile or seasonal series	Moving average

Tips and Tricks

Show your working for at least one calculation of each type. The criterion is about processing, and the marker needs to see that you did it rather than downloading a series that was already processed.

Label every figure completely. Units, period, and the source. "GDP" is not a label; "Real GDP, annual % change, seasonally adjusted, Stats NZ" is.

Do not over-process. Converting a series that is already a percentage into an index of percentages usually makes it harder to read, not easier. Every transformation should have a reason you can state in one sentence.

Be careful with the percentage / percentage point distinction when writing about unemployment, interest rates or inflation, because all three are already percentages. A rise "of 25%" and a rise "of 1 percentage point" describe the same movement from 4% to 5% and sound completely different.

Worked ExampleProcessing a series four ways

An economy publishes the following. All figures are invented, for method only.

Year	Nominal GDP ($bn)	Price index (base Year 1 = 1000)	Population (millions)
Year 1	340	1000	5.00
Year 2	366	1050	5.10
Year 3	398	1120	5.22

(a) Calculate the annual percentage change in nominal GDP for each year. (b) Convert each year's GDP to real terms at Year 1 prices. (c) Calculate real GDP growth for each year. (d) Calculate real GDP per capita for each year, and comment.

Step 1 — (a) Nominal percentage change

$\text{Year 1} \rightarrow \text{Year 2}: \frac{366 - 340}{340} \times 100 = 7.6\%$

$\text{Year 2} \rightarrow \text{Year 3}: \frac{398 - 366}{366} \times 100 = 8.7\%$

Nominal GDP is growing, and accelerating. On its own this looks like a strong economy.

Step 2 — (b) Convert to real terms at Year 1 prices

$\text{real} = \text{nominal} \times \frac{\text{base index}}{\text{current index}}$

Year 1: it is the base year, so real = nominal = $340bn
Year 2: $366 \times \dfrac{1000}{1050} = \$348.6\text{bn}$
Year 3: $398 \times \dfrac{1000}{1120} = \$355.4\text{bn}$

Step 3 — (c) Real GDP growth

$\text{Year 1} \rightarrow \text{Year 2}: \frac{348.6 - 340}{340} \times 100 = 2.5\%$

$\text{Year 2} \rightarrow \text{Year 3}: \frac{355.4 - 348.6}{348.6} \times 100 = 2.0\%$

The picture reverses completely. Nominal growth accelerated from 7.6% to 8.7%; real growth slowed from 2.5% to 2.0%.

The whole of the apparent acceleration — and more — was inflation. The price index rose 5.0% in Year 2 and 6.7% in Year 3.

Step 4 — (d) Real GDP per capita

$\text{per capita} = \frac{\text{real GDP}}{\text{population}}$

Year 1: $340 \div 5.00 = \$68{,}000$
Year 2: $348.6 \div 5.10 = \$68{,}353$
Year 3: $355.4 \div 5.22 = \$68{,}085$

Comment. Real GDP per capita rose slightly in Year 2 and fell in Year 3, from $68,353 to $68,085.

Population grew 2.4% in Year 3 while real GDP grew only 2.0%, so the economy was producing more in total but less per person. The average New Zealander's share of output fell.

Step 5 — What the three processing steps revealed

Measure	Year 2 → Year 3	What it says
Nominal GDP	+8.7%	Looks strong — but confounds price and quantity
Real GDP	+2.0%	Genuine growth, and slowing
Real GDP per capita	−0.4%	The average person is worse off

$\text{Three processing steps, three different conclusions from one dataset}$

Processing data: percentages, indexes and real values

Covers AS91226

What "processing" means

The Achieved criterion says "processing and presenting statistical data ... to show trends".
Processing means doing something to the raw numbers so a trend becomes visible: converting to percentages, building an index, adjusting for inflation or population, or calculating a change.
Copying a Stats NZ table into your report is not processing. You must transform it.

Percentage change

$\% \text{ change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100$

Converts levels into rates of change, which is what most economic trends are about.
Always say what period the percentage covers — over a quarter, or over a year.

Percentage change versus percentage point change

If unemployment goes from 4.0% to 5.0%, that is a rise of 1.0 percentage point, and a 25% increase in the rate.
Both are correct; they mean different things. Say which one you are using.

Index numbers

An index rescales a series so every value is compared to a base period, which is set to 100 or 1000.

$\text{index} = \frac{\text{value in year}}{\text{value in base year}} \times 1000$

Why indexes are useful here

They strip out the units, so two series measured in different things — dollars and people, say — can be plotted on the same graph and compared directly.
That is exactly what the inter-relationship section needs.

Real values

Converting a nominal series to a real one removes the effect of price changes:

$\text{real value} = \text{nominal value} \times \frac{\text{price index in base year}}{\text{price index in current year}}$

Use it for GDP, wages, incomes and government spending — any series measured in dollars over time.
Approximation for rates of change:

$\text{real change} \approx \text{nominal change} - \text{inflation rate}$

Per capita values

$\text{per capita} = \frac{\text{total}}{\text{population}}$

Essential when population is changing. Real GDP can rise 2% while population rises 3%, so real GDP per capita falls — the average person is worse off even though the economy grew.
Any comparison over a long period, or between regions of different sizes, should be per capita.

Averages and moving averages

A moving average smooths a volatile series so the underlying trend shows through.
A four-quarter moving average averages each quarter with the three before it, which removes most seasonal noise.
Useful when the raw series jumps around so much that the trend cannot be seen.

Which processing method for which job

You want to…	Use
Show how fast something is changing	Percentage change
Compare two series measured in different units	Index numbers
Remove the effect of inflation	Real values
Account for a changing population	Per capita
See through a volatile or seasonal series	Moving average

Tips and Tricks

Label every figure completely. Units, period, and the source. "GDP" is not a label; "Real GDP, annual % change, seasonally adjusted, Stats NZ" is.

Worked ExampleProcessing a series four ways

An economy publishes the following. All figures are invented, for method only.

Year	Nominal GDP ($bn)	Price index (base Year 1 = 1000)	Population (millions)
Year 1	340	1000	5.00
Year 2	366	1050	5.10
Year 3	398	1120	5.22

Step 1 — (a) Nominal percentage change

$\text{Year 1} \rightarrow \text{Year 2}: \frac{366 - 340}{340} \times 100 = 7.6\%$

$\text{Year 2} \rightarrow \text{Year 3}: \frac{398 - 366}{366} \times 100 = 8.7\%$

Nominal GDP is growing, and accelerating. On its own this looks like a strong economy.

Step 2 — (b) Convert to real terms at Year 1 prices

$\text{real} = \text{nominal} \times \frac{\text{base index}}{\text{current index}}$

Year 1: it is the base year, so real = nominal = $340bn
Year 2: $366 \times \dfrac{1000}{1050} = \$348.6\text{bn}$
Year 3: $398 \times \dfrac{1000}{1120} = \$355.4\text{bn}$

Step 3 — (c) Real GDP growth

$\text{Year 1} \rightarrow \text{Year 2}: \frac{348.6 - 340}{340} \times 100 = 2.5\%$

$\text{Year 2} \rightarrow \text{Year 3}: \frac{355.4 - 348.6}{348.6} \times 100 = 2.0\%$

The picture reverses completely. Nominal growth accelerated from 7.6% to 8.7%; real growth slowed from 2.5% to 2.0%.

The whole of the apparent acceleration — and more — was inflation. The price index rose 5.0% in Year 2 and 6.7% in Year 3.

Step 4 — (d) Real GDP per capita

$\text{per capita} = \frac{\text{real GDP}}{\text{population}}$

Year 1: $340 \div 5.00 = \$68{,}000$
Year 2: $348.6 \div 5.10 = \$68{,}353$
Year 3: $355.4 \div 5.22 = \$68{,}085$

Comment. Real GDP per capita rose slightly in Year 2 and fell in Year 3, from $68,353 to $68,085.

Population grew 2.4% in Year 3 while real GDP grew only 2.0%, so the economy was producing more in total but less per person. The average New Zealander's share of output fell.

Step 5 — What the three processing steps revealed

Measure	Year 2 → Year 3	What it says
Nominal GDP	+8.7%	Looks strong — but confounds price and quantity
Real GDP	+2.0%	Genuine growth, and slowing
Real GDP per capita	−0.4%	The average person is worse off

$\text{Three processing steps, three different conclusions from one dataset}$