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: ${\frac {50}{100}}\times {\frac {40}{100}}$ = $0.50\times 0.40$ = $0.20$ = ${\frac {20}{100}}$ = $20\%$

Averages

An average is simply a number that is representative of data. More particularly, it is a measure of central tendency. There are several types of average. Averages are useful for comparing data, especially when sets of different size are being compared. It acts as a representative figure of the whole set of data.

Perhaps the simplest and commonly used average the arithmetic mean or more simply mean which is explained in the next section.

Other common types of average are the median, the mode, the geometric mean, and the harmonic mean, each of which may be the most appropriate one to use under different circumstances.

Mean, Median and Mode

Mean

The mean, or more precisely the arithmetic mean, is simply the arithmetic average of a group of numbers (or data set) and is shown using -bar symbol ${\bar {}}$. So the mean of the variable $x$ is ${\bar {x}}$, pronounced "x-bar". It is calculated by adding up all of the values in a data set and dividing by the number of values in that data set :${\bar {x}}={\sum _{}x \over n}$.For example, take the following set of data: {1,2,3,4,5}. The mean of this data would be:

${\bar {x}}={\sum _{}x \over n}={1+2+3+4+5 \over 5}={15 \over 5}=3$

Here is a more complicated data set: {10,14,86,2,68,99,1}. The mean would be calculated like this:

${\bar {x}}={\sum _{}x \over n}={10+14+86+2+68+99+1 \over 7}={280 \over 7}=40$

Median

The median is the "middle value" in a set. That is, the median is the number in the center of a data set that has been ordered sequentially.

For example, let's look at the data in our second data set from above: {10,14,86,2,68,99,1}. What is its median?

• First, we sort our data set sequentially: {1,2,10,14,68,85,99}
• Next, we determine the total number of points in our data set (in this case, 7.)
• Finally, we determine the central position of or data set (in this case, the 4th position), and the number in the central position is our median - {1,2,10,14,68,85,99}, making 14 our median.

Because our data set had an odd number of points, determining the central position was easy - it will have the same number of points before it as after it. But what if our data set has an even number of points?

Let's take the same data set, but add a new number to it: {1,2,10,14,68,85,99,100} What is the median of this set?

When you have an even number of points, you must determine the two central positions of the data set. (See side box for instructions.) So for a set of 8 numbers, we get (8 + 1) / 2 = 9 / 2 = 4 1/2, which has 4 and 5 on either side.

Looking at our dataset, we see that the 4th and 5th numbers are 14 and 68. From there, we return to our trusty friend the mean to determine the median. (14 + 68) / 2 = 82 / 2 = 41. find the median of 2, 4, 6, 8 => firstly we must count the numbers to determine its odd or even as we see it is even so we can write: M=(4+6)/2=10/2=5 5 is the median of above sequential numbers.

Mode

The mode is the most common or "most frequent" value in a data set. Example: the mode of the following data set (1, 2, 5, 5, 6, 3) is 5 since it appears twice. This is the most common value of the data set. Data sets having one mode are said to be unimodal, with two are said to be bimodal and with more than two are said to be multimodal . An example of a unimodal dataset is {1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 9}. The mode for this data set is 4. An example of a bimodal data set is {1, 2, 2, 3, 3}. This is because both 2 and 3 are modes. Please note: If all points in a data set occur with equal frequency, it is equally accurate to describe the data set as having many modes or no mode.

Midrange

The midrange is the arithmetic mean strictly between the minimum and the maximum value in a data set.

Relationship of the Mean, Median, and Mode

The relationship of the mean, median, and mode to each other can provide some information about the relative shape of the data distribution. If the mean, median, and mode are approximately equal to each other, the distribution can be assumed to be approximately symmetrical. If the mean > median > mode, the distribution will be skewed to the right. If the mean < median < mode, the distribution will be skewed to the left.

Licensing

Content obtained and/or adapted from:

• Ratio, Wikipedia under a CC BY-SA license
• Fraction, Wikipedia under a CC BY-SA license
• Statistics/Summary/Averages/Mean, Wikibooks under a CC BY-SA license