Showing posts with label bell curve. Show all posts
Showing posts with label bell curve. Show all posts

## Monday, March 29, 2021

### Normal Distribution or Bell Curve

The bell curve is also known as the normal curve or normal distribution. The bell curve has mathematical properties that allow researchers to draw conclusions about where scores (or data) are located relative to other scores (or data).

The three measures of central tendency (mode, median, mean) are at the same middle point in a normal curve. The numbers representing the middle of the bell curve divide the distribution in half.

On the x-axis in the normal distribution, the mean is at zero and there are standard deviation units above and below the mean.

The height of the curve indicates the percentage of scores in that area. You can see that a large percentage of the scores are between 1 and -1 standard deviations. About 68% of scores fall between +1 and -1 standard deviations from the mean.

Look at the illustration below to see that there are about 34% of the scores in falling one standard deviation above the mean and another 34% in one standard deviation below the mean. Thus, 34% + 34% = 68%.

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Example: Intelligence test scores (IQ) appear to be distributed like the normal distribution within each age group. The mean IQ is 100 and one standard deviation is 15 points on most IQ tests. Thus, 68% of people of a similar age have IQ scores between 85 and 115.

Knowing that 100 is the average or mean IQ then we know that half of people taking the test are below average intelligence (as measured by the test) and half are above average intelligence.

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The ends of the normal distribution are called tails. Extreme scores are in the tails. Consider the low level of the the distribution at - 2.5 or +2.5 standard deviations. At these points, the curve almost touches the x-axis; however, it never the lines of the curve never quite touch the x-axis.

Only a small percentage of scores is beyond 2.5 standard deviations in either direction. Theoretically, the tails of the curve never touch the baseline. Only a small fraction of a percent of scores is beyond 3 standard deviations.

The picture below illustrates the percentage of scores (or data) within different areas of the curve. For example, on a normally distributed test, 34.1% of scores will fall between the mean and 1 standard deviation above the mean. Because the curve is symmetrical, the same percentage will be between the mean and 1 standard deviation below the mean.

Applied Statistics Concepts for Counselors at or    GOOGLE

Related posts/ pages

A-Z list of statistics

Skewed distributions

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## Sunday, September 3, 2017

### Reporting Mean or Median

Who would think that a simple statistic like a mean or a median would make a difference?

In large samples involving thousands of people, and when data are normally distributed (close to the shape of a bell curve), the mean and median will be nearly the same. In fact, in a theoretical distribution called the normal curve, the mean, median, and  mode are in the middle.

But, many samples are not normal distributions. Instead, the often contain extreme scores called outliers or a lot of scores bunched up at high or low levels (skewed). Sadly, even people that understand statistics, continue to report the mean as if they are not thinking about their samples.

Suppose you work for a company where the top person earns \$300,000 but most folks earn \$30,000 to \$60,000. Well that \$300,000 is gonna skew results and the mean will look much higher than the median.

I ran some fictitious data on a sample of 10 people. Nine earn between \$30 and \$60K and one earns \$300K. The Mean = \$67K (standard deviation = 82.58), but the Median is only \$38.5K and the Range = \$270K.

Now those results are fictitious and it is a small sample so it magnifies the differences. But you know some folks are earning over \$1,000,000.00 in some companies and lots of folks aren't earning anywhere near that amount.

So who cares? Well salaries make a lot of difference if you are arguing for a raise, considering a change of jobs, voting on budgets in not-for-profit organizations, and more. How motivating is it to give a donation to a company that helps the poor where the CEO pulls down nearly a million bucks a year and you get by on \$65K-- or less?

But there's more. Teacher evaluations are usually skewed -- most students give high ratings-- so the median and range are more appropriate than the mean.

Real estate prices can be out-of-whack if you look at the mean price in a city where a few multimillion dollar homes pull the mean to a high level compared to the median price.

I see research papers where the scientists report the average age of people in surveys is 19 and they tell you thir sample was from a university. No problem with age 19 but when they report a Mean of 19 and a standard deviation of 5, there is a problem! If you understand standard deviations, you will know why they probably did not have a lot of 14-year olds in their university!

You can see that knowledgeable folks can play games with a simple statistic.

If you forgot about the meaning of some terms, here's a link to a free glossary.

A simple example

Counselors, teachers, and parents - think about test scores and how they are reported.  Test scores for students at school may be distorted by a few very high scoring or very low scoring students.

"Averages" can be deceiving.

Read more about basic statistics in APPLIED STATISTICS: CONCEPTS FOR COUNSELORS at

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