Math

Wednesday 4/2/2009

Formulae, functions and simple calcucations. This information is adapted from Children's Defense Fund

1. Calculating a Rate

A Rate is simply the number of things per some other number, usually 100 or 1,000 or some other multiple of 10. A percentage is a rate per 100. Infant mortality rates are calculated per 1,000. To calculate a rate, you need three pieces of information:

• The number in the total group (for example, the total number of babies born in a given year);
• The number in the subgroup you are interested in (such as the number of infants that died that year); and
• The per number (e.g., per 100, or 1,000, or 100,000, etc). The per number is called a multiplier.

The formula for calculating a rate is:

(Number in subgroup ÷ Number in total group) × multiplier

In 2003, 135,979 babies were born in Planet-X. In that same year, 1,151 infants died. The infant mortality rate is the number of infant deaths per 1,000 births. Calculate Planet-X's 2003 infant mortality rate this way:

(1,151 ÷ 135,979) × 1,000 = 0.00846 (rounded) × 1,000 = 8.46

To calculate the rate, be sure to work with numbers that are large enough to be meaningful. A general rule is: if the number of people or events or things is less than 30, do not calculate a rate. Rates based on very small numbers can vary hugely from year to year and are not stable.

2. Calculating a Ratio

A ratio is one number divided by another. A ratio tells you how much bigger or smaller one number is compared to the other. For example, in 2003 the infant mortality rate among group-A was 14.01 whilst the rate among group-B was 5.72. The ratio of the group-A rate to the group-B rate is:

14.01 ÷ 5.72 = 2.449

Any two numbers can be compared in this way, provided there exists the same measure for the two groups in the same year (as in the infant mortality example above) or, for one group in two different years (such as the unemployment rate in 2004 compared with the rate in 2000).

3. Calculating Change Over Time

If data are known for two or more points in time, then it's possible to calculate how much change there was between the first and second points. Typically, knowing the size of the change (the percent by which the number changed) is important. Two pieces of information are required to calculate a rate of change over time:

• The number from the earlier point in time, and
• The number from the later point in time.

The rate of change is:

[(Number at later time ÷ Number at earlier time) - 1] × 100

If the resulting number is positive, then this means there was an increase over time. Conversely, if the result is negative, then this means there was a decrease. For example, if an infant mortality rate was 8.4 in 1993 and 5.8 in 2003, then calculate the rate of change this way:

   5.8 ÷ 8.4 =  0.69048
 0.69048 - 1 = -0.30952
-0.30952 × 100 = -30.95%

Which means that the infant mortality rate decreased by 31.0% between 1993 and 2003

4. Ranking

Ranking permits observations between objects or groups. E.g., the statement There are 17 other Planet's with lower infant mortality than Planet-X expresses a ranking. Ranking should only be done when it is possible to compare the same measure of data for each of the groups being observed.

Ranking involves:

• Putting the numbers in order and, then;
• Assigning a rank to each based on its place in the order

Numbers can be ordered from largest to smallest or from smallest to largest. To rank numbers, order them from best to worst. The best number can be assigned Rank 1, with Rank 2 going to the second best, and so forth down the list. The worst number are assigned the lowest rank. Best and worst are determined by the specific number. The largest value would be best, for example, when compared against median income, whereas the smallest value would be best for infant mortality. When two or more numbers in the list are the same, it does not make sense to give them different ranks. Give them the same rank and skip the next rank. If three numbers are the same, give them the same rank and skip the next two rankings.

5. Standard Deviation

The standard deviation is the average distance that a given measurement is from the mean of a sample. Put another way, the standard deviation is defined as the average amount by which values in a distribution differ from the mean, ignoring the sign of the difference. For example, in a sample of 1,2,3,4,5,6,7,8,9, the mean is (1+2+3+4+5+6+7+8+9) / 9 = 5 and a value such as 2 is 3 units below the average (calculated as 5 - 2 = 3).

To calculate a standard deviation, first establish the mean (which is the sum of a set of numbers divided by the number of elements in that set of numbers). For each number in the list, subtract off the calculated mean and then square the result. E.g., for the numbers in the sample set 1,2,3,4,5,6,7,8,9 the caluclation is:

(1-5)² = 16
(2-5)² =  9
(3-5)² =  4
(4-5)² =  1
(5-5)² =  0
(6-5)² =  1
(7-5)² =  4
(8-5)² =  9
(9-5)² = 16

Next, sum the previously calculated squares and divide the result by 1 less than the number of elements in the the sample set (i.e., by 8 in the previous sample). The standard deviation will be the square root of this calculation. E.g.,

 sqrt([16+9+4+1+0+1+4+9+16] / 8) = 2.738612788

Repeating this in steps:

• set of numbers: 1,2,3,4,5,6,7,8,9
• mean: (1+2+3+4+5+6+7+8+9) / 9 = 5
• set of deviations: -4, -3, -2, -1, 0, 1, 2, 3, 4
• squares of deviations: 16, 9, 4, 1, 0, 1, 4, 9, 16
• sum of deviations: 16 + 9 + 4 + 1 + 0 + 1 + 4 + 9 + 16 = 60
• divided by one less than the number of items in the set: 60 / 8 = 7.5
• square root of previous number: '''

5.1. Characteristics of a standard deviation

The mean (x̄) of a sample and its standard deviation (s) determine the shape of the data's distribution. The highest point on the curve is the average and the distribution is symmetrical about the average. 99.7% of the area under the curve lies between -3s and +3s of the average (x̄). 95.44% of the curve is between -2s and +2s of the average, while 68.26% of the curve is between -1s and +1s of the average. A normal distribution looks like this:


From http://www.spcforexcel.com/explaining-standard-deviation#standard-deviation

The standard deviation has the same unit as the data set unit.

See also: http://www.une.edu.au/WebStat/unit_materials/c4_descriptive_statistics/standard_deviation.htm

Stuart Moorfoot 4 Feb 2009 foo@bund.com.au