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- Percentages
- Index numbers and indexes
- Means and medians
Percentages
One of the most frequent ways to represent statistics is by percentage. Percent simply means "per hundred" and the symbol used to express percentage is %. One percent (or 1%) is one hundredth of the total or whole and is therefore calculated by dividing the total or whole number by 100.
Example: 1% of 250 = (1 ÷ 100) x 250 = 2.5
To calculate a given percentage of a number, divide the total number by 100 and then multiply the result by the requested percentage:
Example: 12% of 250 = (250 ÷ 100) x 12 = 30
To calculate what percentage one number is of another number, change this equation around and multiply the first number by 100 and then divide the result by the second number:
Example: 30 as a % of 250 = (30 x 100) ÷ 250 = 12%
To determine a percentage of the total from a series of numbers, add the numbers in the series to find the total (i.e. the number equal to 100%) and carry out the above calculation for each number in the series:
Example: Given the series 30,150,70:
The total would be 30 + 150 + 70 = 250
30 as a % of 250 = (30 x 100) ÷ 250 = 12%
150 as a % of 250 = (150 x 100) ÷ 250 = 60%
70 as a % of 250 = (70 x 100) ÷ 250 = 28%
If the percentages for each number in the series are added together, they equal the percentage for the whole: 12% + 60% + 28% = 100%
To calculate the percentage difference between two numbers, the same basic calculations are used.
Example: To find the percentage change from 250 to 280, the
difference between numbers is calculated:
280 – 250 = 30
and then expressed as a percentage of the first, or base, number:
(30 x 100) ÷ 250 = 12%
To determine the whole number (i.e. the value of 100%) when only the value of a given percentage:
Example: If 280 is known to be 112%
then 1% must be 280 ÷ 112 = 2.5
and 100% must be (280 x 100) ÷ 112 = 250
To compare a number of different things, they need to be expressed on the same base:
Example: if the price of sausage increased from $2.99 per kilogram to $3.99 and the same quantity of wieners from $1.99 to $2.99, the two increases could be expressed as percentages.
Sausages: $3.99 - $2.99 = $1.00
$1.00 as a % of $2.99 is ($1.00 x 100) ÷ $2.99 = 33%
Wieners: $2.99 – $1.99 = $1.00
$1.00 as a % of $1.99 is ($1.00 x 100) ÷ $1.99 = 50%
It is now easy to see that the price increase of wieners was much higher than that of sausages.
It should be remembered that comparing percentages which have significantly different bases can create a false impression.
Example: The change from one to two is 100% whereas the change from 5,000,000 to 6,000,000 is only 20%.
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Index numbers and indexes
Index numbers are a statistician's way of expressing the difference between two measurements by designating one number as the "base", giving it the value 100 and then expressing the second number as a percentage of the first.
Example: If the population of a town increased from 20,000 in 1988 to 21,000 in 1991, the population in 1991 was 105% of the population in 1988. Therefore, on a 1988 = 100 base, the population index for the town was 105 in 1991.
An "index", as the term is generally used when referring to statistics, is a series of index numbers expressing a series of numbers as percentages of a single number.
Example: the numbers
507590110
expressed as an index, with the first number as a base, would be
100150180220
Indexes can be used to express comparisons between places, industries, etc. but the most common use is to express changes over a period of time, in which case the index is also a time series or "series". One point in time is designated the base period—it may be a year, month, or any other period—and given the value 100. The index numbers for the measurement (price, quantity, value, etc.) at all other points in time indicate the percentage change from the base period.
If the price, quantity or value has increased by 15% since the base period, the index is 115; if it has fallen 5%, the index is 95. It is important to note that indexes reflect percentage differences relative to the base year and not absolute levels. If the price index for one item is 110 and for another is 105, it means the price of the first has increased twice as much as the price of the second. It does not mean that the first item is more expensive than the second.
Each index number in a series reflects the percentage change from the base period. It is important not to confuse an index point change and a percentage change between two index numbers in a series.
Example: if the price index for butter was 130 one year and 143 the next year, the index point change would be:
143 – 130 = 13
but the percentage change for the index would be:
(143 – 130) x 100) ÷ 130 = 10%
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Means and medians
These are both ways of expressing a series of numbers by a single number. The mean most frequently referred to in Statistics Canada's publications is the arithmetic mean. It is what most people call the "average" and is calculated by adding up the numbers in the series and dividing the total by however many numbers there are.
Example: If five children are aged respectively 3, 4, 5, 8 and 10 years old, their mean age is:
3 + 4 + 5 + 8 + 10 = 6
5
The median is the value of the middle number of a series ranked in order of size.
Example: Given the ages of five children as 5, 4, 8, 3 and 10, to find the median age the series would first have to be rearranged in order of size, i.e. 3, 4, 5, 8, 10 and the value of the middle number, i.e. 5, would be the median age.
