What is the highest point on a bell curve called?

Scores used in statistics can be hard to interpret, but one of the basic ways that statistics describes academic scores is with the bell curve, also known as the normal distribution or Gaussian distribution. Understanding this curve and how scores fall on it can make statistics much easier to interpret and understand. You may see T-scores, Z-scores, standard scores or even stanines reported. One thing they all have in common is that they are scores distributed on the same bell curve. The bell curve and its properties never change. The only thing that changes is a specific score and where it would fall on the bell curve. If you ever read a report with a score on it, make sure you find out the type of score it is. Once you know that, you should be able to look at the bell curve to see what the score really means.

Look at the symmetrical shape of a bell curve. The center should be where the largest portion of scores would fall. The smallest areas to the far left and right would be where the very lowest and very highest scores would fall.

Read across the curve from left to right. The curve is typically broken down into sections. Each section represents the portion, or percentage, of scores that would fall at that point on the curve. The first, or smallest, section might only represent a few scores. The largest portion of the scores would be in the two sections nearest the center, where 68.26 percent of the scores would fall. All the percentages for the different sections add up to 100 percent, with 50 percent falling on each side of the curve. The left of the curve represents scores that fall below the average and the right side represents scores that fall above the average.

Look for a line labeled "standard deviations." The standard deviation is the key to interpreting scores that fall on the bell curve. The standard deviation is how many scores are disbursed in that section of the curve. Different types of scores have different standard deviations. For example, a standard score usually has a standard deviation of 15, and a T-score always has a standard deviation of 10.

Find out the type of score your are looking at. A score may seem good, but you have to know what type of score it is to really know. For example, people are used to 100 being a good score because that stands for a perfect score in school. A score of 60 then would be considered a bad score. If that 60 is a T-score, however, it is above average for whatever it is measuring.

Read down the side of the bell curve to find the types of scores. Look across the line for that type of score. The T-score that falls on the mean is 50, while the z-score is zero. Many scores that are reported are called "standard scores." Standard scores have an average of 100. So a standard score of 100, a T-score of 50 and a Z-score of 0 all mean the same thing because they all fall at the same point on the bell curve. Another way of putting it is that a standard score of 100 would convert to a T-score of 50.

By Dr. Saul McLeod, published 2019

What are the properties of the normal distribution?

The normal distribution is a continuous probability distribution that is symmetrical on both sides of the mean, so the right side of the center is a mirror image of the left side.

The area under the normal distribution curve represents probability and the total area under the curve sums to one.

Most of the continuous data values in a normal distribution tend to cluster around the mean, and the further a value is from the mean, the less likely it is to occur. The tails are asymptotic, which means that they approach but never quite meet the horizon (i.e. x-axis).

For a perfectly normal distribution the mean, median and mode will be the same value, visually represented by the peak of the curve.

The normal distribution is often called the bell curve because the graph of its probability density looks like a bell. It is also known as called Gaussian distribution, after the German mathematician Carl Gauss who first described it.

What is the difference between a normal distribution and a standard normal distribution?

A normal distribution is determined by two parameters the mean and the variance. A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution.

  Figure 1. A standard normal distribution (SND).

This is the distribution that is used to construct tables of the normal distribution.

Why is the normal distribution important?

The bell-shaped curve is a common feature of nature and psychology

The normal distribution is the most important probability distribution in statistics because many continuous data in nature and psychology displays this bell-shaped curve when compiled and graphed.

For example, if we randomly sampled 100 individuals we would expect to see a normal distribution frequency curve for many continuous variables, such as IQ, height, weight and blood pressure.

Parametric significance tests require a normal distribution of the samples' data points

The most powerful (parametric) statistical tests used by psychologists require data to be normally distributed. If the data does not resemble a bell curve researchers may have to use a less powerful type of statistical test, called non-parametric statistics.

Converting the raw scores of a normal distribution to z-scores

We can standardized the values (raw scores) of a normal distribution by converting them into z-scores.

This procedure allows researchers to determine the proportion of the values that fall within a specified number of standard deviations from the mean (i.e. calculate the empirical rule).

Probability and the normal curve: What is the empirical rule formula?

The empirical rule in statistics allows researchers to determine the proportion of values that fall within certain distances from the mean. The empirical rule is often referred to as the three-sigma rule or the 68-95-99.7 rule.

If the data values in a normal distribution are converted to standard score (z-score) in a standard normal distribution the empirical rule describes the percentage of the data that fall within specific numbers of standard deviations (σ) from the mean (μ) for bell-shaped curves.

The empirical rule allows researchers to calculate the probability of randomly obtaining a score from a normal distribution.

68% of data falls within the first standard deviation from the mean. This means there is a 68% probability of randomly selecting a score between -1 and +1 standard deviations from the mean.

95% of the values fall within two standard deviations from the mean. This means there is a 95% probability of randomly selecting a score between -2 and +2 standard deviations from the mean.

99.7% of data will fall within three standard deviations from the mean. This means there is a 99.7% probability of randomly selecting a score between -3 and +3 standard deviations from the mean.

How can I check if my data follows a normal distribution?

Statistical software (such as SPSS) can be used to check if your dataset is normally distributed by calculating the three measures of central tendency. If the mean, median and mode are very similar values there is a good chance that the data follows a bell-shaped distribution (SPSS command here).

It is also advisable to a frequency graph too, so you can check the visual shape of your data (If your chart is a histogram, you can add a distribution curve using SPSS: From the menus choose: Elements > Show Distribution Curve).

Normal distributions become more apparent (i.e. perfect) the finer the level of measurement and the larger the sample from a population.

You can also calculate coefficients which tell us about the size of the distribution tails in relation to the bump in the middle of the bell curve. For example, Kolmogorov Smirnov and Shapiro-Wilk tests can be calculated using SPSS.

These tests compare your data to a normal distribution and provide a p-value, which if significant (p < .05) indicates your data is different to a normal distribution (thus, on this occasion we do not want a significant result and need a p-value higher than 0.05).

How to reference this article:

McLeod, S. A. (2019, May 28). Introduction to the normal distribution (bell curve). Simply psychology: https://www.simplypsychology.org/normal-distribution.html

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