Normal Distribution Psychology

Normal Distribution: Psychology Definition

Normal distributions are bell-shaped, symmetrical graphical illustrations where values such as the mean, median, and mode sit at the centre of the bell, and extreme scores tail off at either end. An IQ (intelligence) test is a classic example of a normal distribution in psychology.

Most people tend to have an IQ score between 85 and 115, and the scores are normally distributed. Height, shoe size or personality traits like extraversion or neuroticism tend to be normally distributed in a population.

Fig. 1 - Normal distributions are characterised by their notable bell shapes¹.

Features of Normal Distributions in Psychology

Normal distributions have distinct features. They are symmetrical, meaning that the distribution of scores larger than the mean should be symmetrical to the distribution of scores smaller than the mean. Normally distributed data forms a bell curve. The mean, median and mode values are similar or the same, creating the distribution's centre (the peak).

The distribution's tails meet the x-axis at infinity, meaning they shouldn't touch the x-axis (asymptotic). An equal amount of extreme scores should fall on both sides of the mean.

Normal Distribution Bell Curve in Psychology

The normal distribution can be illustrated with a bell curve in that it's shaped like a bell, which suggests that most of our data is clustered around the mean value in the distribution centre. The probability of values being close to the mean is much higher than those far from the mean because the frequency of scores at the distribution's tails is much lower than in the centre.

The Mean, Median, and Mode in a Normal Distribution

When the data is normally distributed, the mean, median, and mode tend to be the same or similar. Let's recap what each value indicates and how to calculate them.

• The mean is the average value of our data. To calculate the mean, add all the values collected and divide their sum by the number of data points (e.g. the number of participants or measurements).
• The median is the middle value in your data set. Order your data from the lowest to the highest value to find the median. If your number of data points is odd (e.g. 7), choose the value in the middle (the fourth value); if it's even (e.g. 8), then your median is the average of the two middle values (the average of the fourth and the fifth value).
• The mode is the most frequently repeating value in your data set. To find the mode, order your data from the lowest to the highest; the number appearing most frequently is your mode. If none of your values repeats, your data set has no mode.

Normal and Skewed Distributions in Psychology

One of the main features of the normal distribution is symmetry and characteristic bell shape. What happens if our mean, median and mode are different and the distribution isn't symmetrical? In this case, we can identify negatively or positively skewed distribution.

Skewed Distribution

Skewed distributions are not symmetrical; they have their peak off-centre and extended tails in one direction. In the case of skewed distributions, the mean is no longer in the centre. It is shifted due to a large number of extreme scores to one side of the distribution. Such distributions no longer form a bell curve.

Examples of Normal and Skewed distributions

If we know that our mean, median and mode are the same or similar, we can estimate that the data is normally distributed around these values.

Fig. 2 - A normal distribution.

The distribution won't be normal if there is a bigger difference between the mean, median and mode. Here, the mode and median are higher than the mean, suggesting that negatively skewed our data.

Fig. 3 - A negatively skewed distribution.

In this case, our measures of central tendency are quite different; the mode and the median are lower than the mean, suggesting a positively skewed distribution.

Fig. 4 - A positively skewed distribution.

Normal Distribution in Psychological Testing

A normal distribution is quite important in psychological research, especially when making predictions. Here are some examples of normal distributions in psychological testing.

Clinical Practice

The normal distribution is often used in psychological testing when interpreting test scores. As many psychological traits or symptoms are normally distributed across the population, by looking at where an individual score falls on the distribution, we can get an idea of how much it deviates from the average, which can aid diagnosis or help identify people at risk.

Research

Normal distributions are also crucial for research and statistical testing. Inferential statistical tests (e.g. the t-test) that we use to decide whether to reject our null hypothesis mostly require the data to be normally distributed. We will need to use less sensitive, nonparametric statistical tests if our data is not normally distributed.