Fractions and Percentages (Psychology)
Introduction
A fraction is a proportion and is usually not a whole number. The form of a fraction is $\bigg(\dfrac{\text{numerator}}{\text{denominator}}\bigg)$. Fractions and percentages are useful in psychology for displaying data and are used in data analysis.
To simplify fractions you need to look for a common factor of the numerator and denominator and divide both by this factor. Repeat this process until they have no more factors in common (they are then said to be coprime). For example, to simplify $\frac{48}{18}$ we see $48$ and $18$ have a common factor of $6$, so divide by $6$ to get the simplified fraction of $\frac{8}{3}$.
See also fractions.
A percentage is a proportion of an amount, group or set and is out of $100$. Another way to think of this is as a fraction out of $100$. So if you had an $80$% success rate, this would mean that for every $100$ attempts $80$ of them would be successful.
See also percentages
Converting Fractions to Percentages
To convert fractions to percentages divide the numerator (number on the top) by the denominator (number on the bottom) and multiply by $100$. This will give you the fraction as a percentage.
Worked Example
Convert $\dfrac{17}{20}$ into a percentage.
Calculating percentage changes
You calculate a percentage change when the amount of something you have changes. Use the following methods when you know the original value and the new value and you want to calculate the percentage change.
For example, you can calculate the percentage change of children diagnosed with ADHD in a school over a period of time.
Percentage increase:
If the amount you have increases, we use the formula:
\begin{equation} \frac{\text{(new value} - \text{original value)}}{\text{original value}}\times100 = \text{ Percentage increase}. \end{equation}
Percentage decrease:
If the amount you have decreases, we manipulate the above formula to stop it being negative by swapping the two values in the numerator of the fraction:
\begin{equation} \frac{\text{(original value} - \text{new value )}}{\text{original value}}\times100 = \text {Percentage decrease}. \end{equation}
Using percentage change to calculate new amounts:
This method is used when you know the percentage change and the original value and you want to calculate how much you now have. To do this use the formula:
\begin{equation} \frac{\text{(new percentage)}}{100}\times\text{(original value)} = \text{New amount}. \end{equation}
' Note:' These formulas can be rearranged to suit different questions.
Important Note
The examples covered on this page are purely hypothetical and any results or data are not from any real life cases. Their purpose is to demonstrate how to use the various mathematical techniques covered in this section.
Worked Examples
For each of the cases above we will work though an example.
Solution
Firstly calculate the average test score from each column, you do this using the mean of the data.
The mean of the test scores before 'active learning' was introduced : $13$.
The mean of the test scores after 'active learning' was introduced: $17.3$.
There has been an increase in test scores, now to calculate the percentage increase we use the above formula:
\[\bigg(\dfrac{~\text{new value - original value}~}{~\text{original value}~}\bigg)\times 100\]
to give:
\[\bigg(\dfrac{17.3 - 13}{13}\bigg)\times100 \approx 33.1\%.\] So there is a 33.1$\%$ percentage increase in test scores.
Worked Example - Percentage Decrease
Below is a table of men and women diagnosed with depression over the last 6 months. If the number of women diagnosed with depression continues to decrease at the same percentage rate, from June to July and from July to August, as it has from May to June, how many cases of depression in women do you expect to be diagnosed in August?
Gender | Jan | Feb | Mar | Apr | May | Jun |
|---|---|---|---|---|---|---|
Male | 50 | 52 | 61 | 54 | 49 | 47 |
Female | 61 | 72 | 75 | 68 | 67 | 59 |
Solution
Firstly, work out the percentage decrease from May to June. From the above formula, this is:
$\dfrac{67 - 59}{67}\times100 = 11.94$% So, there is a $11.94$% decrease in diagnosed cases of depression in women from May to June. Hence, the amount of women diagnosed with depression in June is $100- 11.94= 88.06$% of those diagnosed in May.
There are two ways in which we can calculate how many cases of depression in women will be diagnosed with depression by August:
1. You can calculate $88.06$% of $59$ to get the expected number for July and then repeat this operation on the expected number for July to get the expected number of cases of depression in August.
2. Or multiply $59$ by $(0.8806)^2$ to get the the number of women expected to be diagnosed in August. Note: you take $0.8806$ to the power 2 as there are two more periods of this rate of decrease.
Both methods are equivalent.
Method 1: $59 \times \dfrac{88.06}{100} = 52$ cases, (rounded to the nearest whole number) in July.
For August, assuming the same percentage decrease:
$52 \times \dfrac{88.06}{100} = 46$ cases, (rounded to the nearest whole number) in August.
Method 2: $59\times(0.8806)^2 = 46$.
So in August you expect to diagnose approximately $46$ women with depression if the current trend continues.
Test Yourself
Test yourself: Background mathematics
Test yourself: Another Numbas test on fractions
External Resourses
- Fractions and percentages from Maths is Fun
