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Normal Distribution or Bell Curve


Understanding the Bell Curve and Standard Deviation

The bell curve—also known as the normal curve or normal distribution—has mathematical properties that allow researchers to understand where scores (or data points) fall in relation to one another. In a normal distribution, most scores cluster around the center, with fewer scores appearing as you move toward the extremes.

Central Tendency in a Normal Curve

In a normal distribution, the three measures of central tendency—mode, median, and mean—all fall at the same central point. This midpoint divides the distribution into two equal halves.
Standard Deviation Units

On the x‑axis of a normal curve, the mean is placed at zero. Standard deviation units extend outward in both directions from the mean. These units help us understand how far a score is from the average.

The height of the curve at any point represents the percentage of scores in that area. A large portion of scores fall close to the mean. In fact, about 68% of all scores lie between –1 and +1 standard deviations.

If you look at the illustration above, you’ll see that roughly 34% of scores fall between the mean and +1 standard deviation, and another 34% fall between the mean and –1 standard deviation. Together, these make up the familiar 68%.

CALCULATOR: Enter Scores at the bottom of the page to obtain basic scores.


Example: IQ Scores

IQ scores are designed to follow a normal distribution within each age group. The average IQ is 100, and the standard deviation on most tests is 15 points. This means that 68% of people score between 85 and 115.

Because 100 is the mean, half of all test‑takers score below 100 and half score above it. This is simply a property of the normal curve—not a judgment about individuals, but a statistical description of how scores tend to fall.

Practice

Learn more by calculating IQ Scores from this set of scores. See the calculator at the bottom of the page.

     Sample set of IQ Scores: 55, 70, 85, 99, 100, 101, 115, 130, 145

In a very large sample, IQ scores and their distance from the mean in SD units would be as follows:

55 (–3 SD)
70 (–2 SD)
85 (–1 SD)
100 (Mean)
115 (+1 SD)
130 (+2 SD)
145 (+3 SD)

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Learn more about test scores in Applied Statistics: Concepts for Counselors available at AMAZON or GOOGLE.

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Understanding the Tails of the Curve

The far ends of the normal distribution are called the tails, where extreme scores appear. At around –2.5 or +2.5 standard deviations, the curve nearly touches the x‑axis—but mathematically, it never actually reaches it.

Only a very small percentage of scores fall beyond ±2.5 standard deviations, and an even smaller fraction—less than one percent—fall beyond ±3 standard deviations. These are truly rare outcomes.
The illustration below shows the percentage of scores within each section of the curve. For example, 34.1% of scores fall between the mean and +1 standard deviation, and because the curve is symmetrical, another 34.1% fall between the mean and –1 standard deviation.

Read more about distributions in Chapter 10 of

Applied Statistics Concepts for Counselors at AMAZON    or    GOOGLE








Related posts/ pages

A-Z list of statistics

Read more in Creating Surveys available on AMAZON and GOOGLE



 





Post Author

Geoffrey W. Sutton, Professor Emeritus of Psychology at Evangel University, holds a master’s degree in counseling and a PhD in psychology from the University of Missouri-Columbia. His postdoctoral work encompassed education and supervision in forensic and neuropsychology and psychopharmacology. As a licensed psychologist, he conducted clinical and neuropsychological evaluations and provided psychotherapy for patients in various settings, including schools, hospitals, and private offices. During his tenure as a professor, Dr. Sutton taught courses on psychotherapy, assessment, and research. He has authored over one hundred publications, including books, book chapters, and articles in peer-reviewed psychology journals. 

His website is https://suttong.com 

You can find Dr. Sutton's books on   AMAZON    and  GOOGLE. 

Many publications are free to download at ResearchGate   and Academia  

Find chapters and essays on Substack. [ @GeoffreyWSutton ]




Score Calculator: Mean, Median, Range, SD, Skew, Kurtosis, Z, Percentiles

Enter scores separated by commas or spaces (e.g., 85, 90, 95, 100):



Summary Statistics

Standard Scores and Percentile Ranks

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