How To Calculate Percentages From The Total

A percentage is way to express a fraction of 100, so if you have any other fraction, all you have to do is convert it to a decimal fraction and multiply by 100. You then express the result with a percent sign (%).

Percentages come in handy in all scientific fields because they provide a ready-made, easy scale to analyze results. For example, you may find that a sample of water weighing 7,481 grams contains 322 grams of solute. If you convert this to a percentage, it's much easier to compare with related measurements.

A percentage of a measurement, or a series of measurements, can only be meaningful if you can calculate a total from which to derive the percentage. When it comes to a measurable quantity such as weight, for example, you simply measure the total weight, and when you are measuring the fraction of a series of measurements, you need the total number of measurements.

You then express the quantity in question as a fraction of the total, and to make the number more useful, you do two more simple operations. The first is to divide the denominator of the fraction into the numerator to get a decimal fraction, which is one with a base of 10. You then multiply by 100 to get a percentage.

In the example mentioned previously, there are 322 grams of solute in a water solution that weighs 7,481 grams. The fraction of solute is 322/7481, which is a difficult number to interpret. However, dividing the denominator into the numerator produces the decimal fraction 0.043, and multiplying by 100 converts this to 4.3 percent. You could do the second operation just as easily by simply moving the decimal point two places to the right.

Percentages are especially helpful when analyzing a population to determine internal characteristics or preferences. This is common in voting polls and demographic studies and even to determine the popularity of a movie.

Again, the percentage calculator only works if you can calculate the total number of units in the population T. Once you have it, you determine the number that displays one characteristic, for example, liking the movie, and the number that displays another characteristic, such as not liking it. You can add as many variables as you want, such as the number of people who were bored by the movie, the number who want to see it twice and so on.

Assign a variable, such as _x_n_, to each characteristic, and the percentage occurrence of that variable is:

\({x_n\over T}\times 100\)

For example, a hypothetical survey of 243 people reveals that 138 liked the movie (_x₁), 40 said they wanted to see it again (x₂), 44 didn't like it (x₃) and 21 were too bored to care (x₄). The corresponding percentages are x₁ = 56.8 percent, x₂ = 16.5 percent, x₃ = 18.1 percent and x₄_ = 8.6 percent.

Suppose you have sample, and you know that a certain percentage displays a particular characteristic (X percent). If you know the total population of the sample T, you can find the number of instances of that characteristic in the sample by using the following procedure, which essentially reverses the procedure for calculating percentages.

Write the percentage as a fraction of 100. For example, X percent = X/100. Let that be equal to y/T:

\({X\over 100} = {y\over T}\text{ }\y= {T\times X\over100}\)

The result y is the number of units in the population that display the characteristic. In a large sample, the number y may contain a fraction. If the sample consists of discrete units that can't be subdivided, round up or down to the nearest integer.