Index numbers

The ratio of width to height of standard-definition television

In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to the ratio 4∶3). Similarly, the ratio of lemons to oranges is 6∶8 (or 3∶4) and the ratio of oranges to the total amount of fruit is 8∶14 (or 4∶7).

The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive.

A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a∶b", or by giving just the value of their quotient ${\frac {a}{b}}$. Equal quotients correspond to equal ratios.

Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers. When two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate.

Converting between decimals and fractions

To change a common fraction to a decimal, do a long division of the decimal representations of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the answer to the desired accuracy. For example, to change ${\frac {1}{4}}$ to a decimal, divide $1.00$ by $4$ ("$4$ into $1.00$"), to obtain $0.25$. To change ${\frac {1}{3}}$ to a decimal, divide $1.000...$ by $3$ ("$3$ into $1.0000...$"), and stop when the desired accuracy is obtained, e.g., at $4$ decimals with $0.3333$. The fraction ${\frac {1}{4}}$ can be written exactly with two decimal digits, while the fraction ${\frac {1}{3}}$ cannot be written exactly as a decimal with a finite number of digits. To change a decimal to a fraction, write in the denominator a $1$ followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits of the original decimal, just omitting the decimal point. Thus $12.3456={\tfrac {123456}{10000}}.$

Converting between ratios and percents

If a mixture contains substances A, B, C and D in the ratio 5∶9∶4∶2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2 parts of D. As 5+9+4+2=20, the total mixture contains 5/20 of A (5 parts out of 20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by the total and multiply by 100, we have converted to percentages: 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as 25∶45∶20∶10).

If the two or more ratio quantities encompass all of the quantities in a particular situation, it is said that "the whole" contains the sum of the parts: for example, a fruit basket containing two apples and three oranges and no other fruit is made up of two parts apples and three parts oranges. In this case, ${\tfrac {2}{5}}$, or 40% of the whole is apples and ${\tfrac {3}{5}}$, or 60% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion.

The percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, one first computes the ratio ${\frac {50}{1250}}=0.04$, and then multiplies by 100 to obtain 4%. The percent value can also be found by multiplying first instead of later, so in this example, the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%.

To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is: ${\frac {50}{100}}\times {\frac {40}{100}}$ = $0.50\times 0.40$ = $0.20$ = ${\frac {20}{100}}$ = $20\%$

Average, median, and mode

There are three primary measures of central tendency, and a couple less often used measures, which each, in their own way, tell us what a typical value is for a set of data. Generally, when finding the measures of central tendency, one would order the values of the data set from least to greatest.

Mode

The mode is simply the number which occurs most often in a set of numbers. For example, if there are seven 12-year olds in a class, ten 13-year olds, and four 14-year olds, the mode is 13, since there are more 13 year olds than any other age. In elections, the mode is often called the plurality, and the candidate who gets the most votes wins, even if they don't get the majority (over half) of the votes.

Median

The median is the middle value of a set of values. For example, if students scored 81, 84, and 93 on a test; we select the middle value of 84 as the median.

If you have an even number of values, the average of the two middle values is used as the median. For example, the median of 81, 84, 86, and 93 is 85, since that's midway between 84 and 86, the two middle values.

Average

The straight average, or arithmetic mean,(sometimes referred to simply as "average" or "mean"), is the sum of all values divided by the number of values. For example, if students scored 81, 84, and 93 on a test, the average is 86.