Quantitative Methods in Social Science 360-300 • CEGEP, Montreal

Revision sheet: index numbers, rates and constant dollars (360-300)

This sheet covers the most useful chapter of the whole Quantitative Methods course: the one that lets you read an economic news release without being told just anything. It does not repeat the formulas, it gives the eight places where a correct percentage becomes a false sentence.

It is written for Social Science students at CEGEP taking the compulsory course 360-300, and it is also useful in economics and history, wherever amounts from different periods are compared. The corrected exercise set on the same chapter goes through every calculation.

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The thread of the chapter

A percentage only means something relative to its DENOMINATOR and its DATE. Every mistake in the chapter comes from adding rates that multiply, or from comparing dollars of two different years.

This chapter is part of 360-300, Quantitative Methods in Social Science

The essentials

Three ways to say that something has changed

• Change in POINTS: the difference of two percentages. Going from $8$ % to $10$ % is an increase of $2$ POINTS.
• Change in PERCENT: $\dfrac{\text{end}-\text{start}}{\text{start}}$ . The same increase is an increase of $25$ %.
• MULTIPLIER: $1+t$ . It is the only object that composes, because successive rates MULTIPLY.
• To undo a decrease of $t$ , you need an increase of $\dfrac{t}{1-t}$ , always larger than $t$ .

The third bar is below the dotted line of the start: an increase of

10

% followed by a decrease of

10

% does not bring you back to

100

but to

99

On an exam, explicitly writing "percentage points" or "percent" is often worth half a mark on its own.

Index numbers: compare by a quotient, never by a difference

• $I=\dfrac{V}{V_{base}}\times 100$ . An index number is a disguised ratio, so two index numbers are compared by their QUOTIENT.
• Going from index $120$ to index $150$ is an increase of $\dfrac{150}{120}-1=25$ %, and not $30$ %.
• CHANGE OF BASE: divide all the index numbers by the one of the new reference year, then multiply by $100$ . The ratios are unchanged.
• Constant dollars: $V_{constant}=V_{current}\times \dfrac{I_{\text{target year}}}{I_{\text{year of the amount}}}$ .

The salary and the prices of the example brought back to 100 in 2005: two different units become comparable, and it is their quotient, 137.8 / 140, that gives purchasing power.

Always write the reference year next to a constant amount: "in 2005 dollars" is part of the result, not of the commentary.

The two formulas that get written wrong

• Average annual rate over $n$ periods: $t=\left(\dfrac{V_{f}}{V_{i}}\right)^{1/n}-1$ . It is a GEOMETRIC mean, never the total increase divided by $n$ .
• Purchasing power: multiplier of the salary DIVIDED by multiplier of prices. Any salary increase smaller than inflation is a real decrease.
• Unemployment rate $=\dfrac{\text{unemployed}}{\text{labour force}}$ ; employment rate $=\dfrac{\text{employed persons}}{\text{population aged 15 and over}}$ .
• Rule of 70: doubling time $\approx \dfrac{70}{\text{rate in percent}}$ , because $\ln 2\approx 0.693$ .

The two rates do not share the same denominator: unemployment is computed on the middle bar, employment on the bottom bar, the one that includes the inactive.

INFLATION: prices rise. DISINFLATION: they rise more slowly. DEFLATION: they fall, the rate is negative. The three words are not interchangeable.

The mistakes that cost marks

These are the errors I correct most often in session. Each one costs marks on a paper, even when the reasoning behind it is right.

1. Adding two successive changes

2 marks, and the mistake is hunted at every exam

What not to write

"The price rose by $10$ % then fell by $10$ %: it is back to its starting level."

What to write

"The multipliers multiply: $1.10\times 0.90=0.99$ , so the price fell by $1$ % overall."

Why: The second change applies to an amount that has ALREADY changed: the decrease of $10$ % applies to $110$ and not to $100$ . It is the one point of the chapter that comes back in every assignment.

2. Confusing percentage points and percent

1 mark, and the marker removes it without discussion

What not to write

"The unemployment rate goes from $8$ % to $10$ %: it rose by $2$ %."

What to write

"It rose by $2$ PERCENTAGE POINTS, that is by $25$ % in relative terms."

From 8% to 10%: a block of 2 points is added to the four starting blocks, that is 2 percentage points, and an increase of 25%.

Why: The two numbers describe the same fact and are not read the same way. The press often writes the first while letting you believe the second.

3. Dividing the total increase by the number of years

2 marks, and the gap grows with the duration

What not to write

"The salary rose by $37.8$ % in twenty years, that is $1.89$ % per year."

What to write

"The average annual rate is $\left(\dfrac{62000}{45000}\right)^{1/20}-1\approx 1.61$ % per year: it is a geometric mean."

At 1.61% compounded per year (green), 45,000 dollars become 62,000 in twenty years; at 1.89%, the naive division, they would exceed 65,000 (red).

Why: An annual rate applies each year to an amount that has already grown. Dividing the total increase by $n$ ignores compound interest and always overestimates the rate.

4. Comparing two index numbers by their difference

1.5 marks, and often the whole table-reading question

What not to write

"The rent index goes from $120$ to $150$ : rents rose by $30$ %."

What to write

" $\dfrac{150}{120}=1.25$ , so rents rose by $25$ %. An index number is compared by a quotient."

From 120 to 150: a block of 30 is added to four blocks of 30, an increase of 25%. The difference of 30 is not a percentage.

Why: An index number is not a percentage: it is a ratio to a base multiplied by $100$ . Subtracting two ratios gives nothing interpretable.

5. Comparing dollars of two different years

the whole question, and the conclusion is reversed

What not to write

"The salary went from $45{,}000$ to $62{,}000$ dollars: teachers gained $17{,}000$ dollars of purchasing power."

What to write

"In 2005 dollars, the $62{,}000$ dollars of 2025 are worth $62{,}000\times\dfrac{100}{140}\approx 44{,}286$ dollars: purchasing power FELL by $1.6$ %."

The solid line rises by

38

% in twenty years; the dashed line, the same salary brought back to 2005 dollars, ends below its starting point.

Why: A 2005 dollar and a 2025 dollar are not the same unit. Comparing the two raw amounts is like comparing metres and feet.

6. Reading a fall in the unemployment rate as job creation

the whole question on the critique of a release

What not to write

"The unemployment rate falls from $6.0$ % to $5.5$ %: so more people are employed."

What to write

"The unemployment rate has the LABOUR FORCE as its denominator. It also falls when discouraged unemployed people stop looking: you have to look at the EMPLOYMENT rate, whose denominator does not move that way."

Worked example: 5 unemployed people out of 60 stop looking and become inactive. The rate falls from 6.0% to 5.5% without a single extra job.

Why: A person who gives up the search leaves the numerator AND the denominator: the rate falls without a single job having been created. It is the classic blind spot of the monthly figure.

7. Calling a slowdown in inflation deflation

1 mark, and it is a vocabulary question that comes back every semester

What not to write

"Inflation goes from $4$ % to $2$ %: we are in deflation."

What to write

"Prices are still rising, twice as slowly: that is DISINFLATION. Deflation would require a NEGATIVE rate."

The inflation rate goes from 4% to 2% (left): that is disinflation, the price index keeps rising, more slowly (right).

Why: The inflation rate is a rate of change: its slowing down does not reverse the movement. As long as the rate stays positive, the price index keeps rising.

8. Comparing crude rates between two populations with different structures

2 marks, and the conclusion of the question

What not to write

"Region A has a death rate of $12$ per thousand against $8$ in region B: people die more there."

What to write

"Crude rates cannot be compared: region A is much older. You must apply the SPECIFIC rates of each region to a COMMON reference age structure."

Illustration: same risks by age in the two regions, 2 and 30 per thousand (labels in deaths per 1,000). Region A, which is older, nevertheless shows 12 per thousand against 8.

Why: A crude rate mixes risk and the composition of the population. Direct standardization removes the effect of the age structure and leaves only what you want to compare.

Which method to choose

Which operation does this percentage call for?

Look at what the problem gives, and above all at whether the quantities are already percentages.

If the two quantities compared are themselves percentages → the difference is in POINTS, the relative change in PERCENT: give both

Example: from $8$ % to $10$ %: $2$ points, that is $25$ %
If several changes follow one another over time → MULTIPLY the multipliers $1+t$ , never add the rates

Example: $+10$ % then $-10$ % gives $\times 0.99$
If you know the start, the end and the number of periods → average annual rate $\left(\dfrac{V_{f}}{V_{i}}\right)^{1/n}-1$

Example: $45{,}000$ to $62{,}000$ in 20 years gives $1.61$ % per year
If the values are index numbers → compare by the QUOTIENT of the index numbers, then subtract $1$

Example: from $120$ to $150$ : $+25$ %
If the problem asks for a doubling time → rule of 70: $\dfrac{70}{\text{rate in percent}}$

Example: at $1.7$ % per year, doubling in $41$ years

If no branch applies, it is almost always because the problem mixes two different denominators: saying so is the answer.

Should this amount be deflated?

Look at the DATES of the amounts compared and at what the question is trying to measure.

If two amounts carry different years and the question is about standard of living, purchasing power or real change → DEFLATE: bring all the amounts back to one reference year with the price index

Example: a salary of 2005 compared with a salary of 2025
If the two amounts carry the same year → no correction: the dollars are already comparable

Example: two regions in the same year
If the question is about what was actually paid out or collected → keep CURRENT dollars, and say so

Example: the amount entered in the budget of the year
If you compare a salary and prices over the same period → divide the multiplier of the salary by that of prices: that is purchasing power

Example: salary $\times 1.378$ and prices $\times 1.40$ give $0.984$

A constant amount is always quoted with its reference year. Without it, the number is comparable to nothing.

How the answer is expected to be written

A marker ticks steps. Here they are in order, with the concluding sentence expected word for word.

Bringing an amount back to constant dollars

When to use it: As soon as a problem compares two amounts from different years, or asks for a "real" change.

1 Choose and ANNOUNCE the reference year, the one whose dollars will serve as the unit.
2 Read off the two price indexes, the one of the year of the amount and the one of the reference year.
3 Apply $V_{constant}=V_{current}\times \dfrac{I_{\text{reference}}}{I_{\text{year of the amount}}}$ .
4 Compare the two amounts now expressed in the same unit, and compute the real change.
5 Conclude by writing the reference year in the sentence.

Concluding sentence

"In constant 2005 dollars, the salary goes from $45{,}000$ to $44{,}286$ dollars: the purchasing power of teachers fell by $1.6$ % in twenty years, despite a nominal increase of $37.8$ %."

The trap: Multiplying instead of dividing: bringing a 2025 amount back to 2005 dollars makes it FALL, since prices have risen in the meantime.

Marking: 1 mark for the announced reference year, 2 marks for the calculation, 1 mark for the real change, 1 mark for the concluding sentence.

Computing and interpreting an average annual rate

When to use it: When the problem gives a starting value, an ending value and a duration.

1 Compute the overall multiplier $\dfrac{V_{f}}{V_{i}}$ and comment on it: above $1$ , it is an increase.
2 Raise this multiplier to the power $\dfrac{1}{n}$ , where $n$ is the number of PERIODS and not the number of dates.
3 Subtract $1$ and convert to a percentage, with two decimals.
4 Check by raising $1+t$ to the power $n$ : you must get back the overall multiplier.

Concluding sentence

"The salary progressed at an average annual rate of $1.61$ %: applied twenty times, this rate does turn $45{,}000$ dollars into $62{,}000$ dollars."

The trap: Counting $21$ periods from 2005 to 2025: there are $21$ dates but $20$ intervals.

Marking: 1 mark for the multiplier, 2 marks for the $n$ th root, 1 mark for the check.

Check before you hand in

Five minutes of checking recover more marks than one more problem started in a hurry.

The direction of deflation

A recent amount brought back to older dollars must DECREASE, since prices have risen. If it increases, the fraction is upside down.

$62{,}000$ dollars of 2025 brought back to 2005 dollars give $44{,}286$ and not $86{,}800$ .
The average annual rate retraces the path

Raise $1+t$ to the power $n$ and compare with the overall multiplier: it is the fastest check of the chapter.

$1.0161^{20}\approx 1.378$ , which matches the announced $+37.8$ %.
The order of magnitude by the rule of 70

At $t$ percent per year, you double in about $\dfrac{70}{t}$ years. It immediately tells you whether an announced rate is plausible.

An average annual rate of $7$ % on a salary would mean doubling in ten years: hardly credible on real data.
Is the denominator the right one?

Before interpreting a rate, write next to it the population used as its denominator, and check that it has not moved between the two dates.

A fall in unemployment accompanied by a fall in the participation rate: the denominator shrank, not unemployment.

The typical problem, taken apart

Twenty years of teacher salary, from nominal to real

The average annual salary of a teacher goes from $45{,}000$ dollars in 2005 to $62{,}000$ dollars in 2025.

The consumer price index, base $100$ in 2005, reaches $140$ in 2025.

We ask for the nominal increase, the average annual rate, the real change in purchasing power and the doubling time of prices.

The data of the problem: the salary in the dollars of each year, and the price index going from 100 to 140.

Step 1

Nominal increase: $\dfrac{62000-45000}{45000}\approx 0.3778$ , that is $+37.8$ %.

Why

It is the figure everyone quotes and the only one that answers no interesting question: it mixes the rise in the salary and the rise in prices.

Step 2

Nominal average annual rate: $\left(\dfrac{62000}{45000}\right)^{1/20}-1\approx 0.0161$ , that is $1.61$ % per year.

Why

Twenty INTERVALS between 2005 and 2025, not twenty-one. The naive division $\dfrac{37.8}{20}=1.89$ % overestimates by nearly three tenths of a point.

Step 3

Inflation: prices are multiplied by $\dfrac{140}{100}=1.40$ , that is $+40$ % in total and $1.40^{1/20}-1\approx 1.70$ % per year.

Why

The two annual rates are now comparable: $1.61$ % against $1.70$ %, and the gap, tiny each year, accumulates over twenty years.

Step 4

Salary of 2025 in 2005 dollars: $62000\times\dfrac{100}{140}\approx 44{,}286$ dollars.

Why

We multiply by the index of the TARGET year over the index of the year of the amount. The inverse would push the salary up to $86{,}800$ dollars, which no plausibility check would let through.

Step 5

Real change: $\dfrac{44286}{45000}-1\approx -0.0159$ , that is $-1.6$ %. Other route: $\dfrac{1.3778}{1.40}\approx 0.9841$ .

Why

The two routes give the same number and that is the check on the calculation: purchasing power is the quotient of the two multipliers, nothing else.

Step 6

Rule of 70: at $1.70$ % per year, prices double in $\dfrac{70}{1.70}\approx 41$ years.

Why

This figure serves as a final check: prices rose by $40$ % in twenty years, so a doubling in about forty years is exactly what we expected.

The conclusion, written out

"In current dollars, the salary progressed by $37.8$ % in twenty years, that is $1.61$ % per year. Since prices rose by $40$ %, that is $1.70$ % per year, the real salary FELL by $1.6$ %: in 2005 dollars, it goes from $45{,}000$ to $44{,}286$ dollars."

The classic mistake on this problem: Concluding on the $17{,}000$ dollar nominal gap: it is the only figure in the problem that has no economic meaning.

Learn by heart

• Successive rates MULTIPLY through their multipliers $1+t$ : $+10$ % then $-10$ % gives $0.99$ .
• Percentage POINTS for the difference of two percentages, PERCENT for the relative change. They are never the same numbers.
• Average annual rate: $\left(\dfrac{V_{f}}{V_{i}}\right)^{1/n}-1$ , and $n$ is the number of INTERVALS.
• Two index numbers are compared by their QUOTIENT: from $120$ to $150$ , it is $+25$ % and not $+30$ %.
• $V_{constant}=V_{current}\times\dfrac{I_{\text{target}}}{I_{\text{of the amount}}}$ , and the reference year is always quoted.
• Purchasing power = multiplier of the salary divided by multiplier of prices. Below $1$ , it is a decrease.
• INFLATION, prices rise. DISINFLATION, they rise more slowly. DEFLATION, they fall.

Frequently asked questions

Why does a 10% increase followed by a 10% decrease not bring you back to the start?

Because the decrease applies to an amount that has already risen. One hundred dollars raised by 10 percent make 110 dollars, and 10 percent of 110 is 11 dollars, so you end up at 99 dollars. Rates do not add: it is the multipliers that multiply, here 1.10 times 0.90, which gives 0.99.

How do you convert an amount to constant dollars?

You multiply the amount by the price index of the reference year, then divide by the index of the year of the amount. A salary of 62,000 dollars in 2025, with an index of 140 base 100 in 2005, is worth 44,286 dollars of 2005. A recent amount always decreases when you bring it back to an earlier year, since prices have risen.

What is the difference between percentage points and percent?

A percentage point is the raw difference between two percentages: from 8 to 10, that is two points. Percent is the relative change: from 8 to 10, that is an increase of 25 percent. Both describe the same fact and give very different numbers, which makes the confusion very useful to anyone who wants to impress.

How do you compute an average annual growth rate?

You divide the final value by the initial value, raise the result to the power one over the number of years, then subtract one. It is never the total increase divided by the number of years, which overestimates the rate because it ignores compounding. You check by raising one plus the rate to the power of the number of years.

Does a fall in the unemployment rate mean more jobs?

Not necessarily. The unemployment rate has the labour force as its denominator, that is, people in work plus those looking for work. When discouraged unemployed people stop looking, they leave both the numerator and the denominator, and the rate falls without a single job being created. The employment rate does not have this flaw.

Practise it

Corrected exercises: Index numbers, rates and constant dollars, 360-300

A method is proved on a paper, not on a sheet. The set for the same chapter takes each of these traps into a problem, with the solution written out step by step.

11 corrected exercises
110 points
150 minutes

Do the exercises

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Revision sheet: index numbers, rates and constant dollars (360-300)

The essentials

Three ways to say that something has changed

Index numbers: compare by a quotient, never by a difference

The two formulas that get written wrong

The mistakes that cost marks

1. Adding two successive changes

2. Confusing percentage points and percent

3. Dividing the total increase by the number of years

4. Comparing two index numbers by their difference

5. Comparing dollars of two different years

6. Reading a fall in the unemployment rate as job creation

7. Calling a slowdown in inflation deflation

8. Comparing crude rates between two populations with different structures

Which method to choose

Which operation does this percentage call for?

Should this amount be deflated?

How the answer is expected to be written

Bringing an amount back to constant dollars

Computing and interpreting an average annual rate

Check before you hand in

The typical problem, taken apart

Twenty years of teacher salary, from nominal to real

Learn by heart

Frequently asked questions

Why does a 10% increase followed by a 10% decrease not bring you back to the start?

How do you convert an amount to constant dollars?

What is the difference between percentage points and percent?

How do you compute an average annual growth rate?

Does a fall in the unemployment rate mean more jobs?

Corrected exercises: Index numbers, rates and constant dollars, 360-300

See also

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