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Practical Applications of the Bell Shaped Curve

Author: Marble Happy
Description:

Define a bell shaped curve and give a little history

Talk about practical uses of the bell shaped curve in different fields

A bell shaped curve should be part of everyone's vocabulary and fundamental knowledge base because the practical applications are endless. By introducing you to the normal distribution (otherwise known as the bell shaped curve), this packet will give you more tools to understand statistics in our everyday world.

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Tutorial

So how DID that happen?

You get an F- on a paper and the teacher stands up in class and announces, "I grade on the curve - deal with it."

What does it mean to "grade on the curve"?

Do you remember standard deviation?  the empirical rule?  the standard normal distribution?

A short review is below:

The standard deviation is a measure of how closely grouped or how widely spaced a set of data appears.  The empirical rule says in a standard normal distribution,  68% of the data points will fall within ± one standard deviation from the mean and 95% will fall within ± two standard deviations.

WAIT!  Did you cover standard normal distribution?  If so, go on, if not, see the following definition:

A standard normal distribution has a mean of 0 and a standard deviation of 1.  Got it?

The normal distribution is important because lots of variables studied in education and psychology are normally distributed.  Interesting stuff like reading ability, job satisfaction and memory, to name a few.  Knowing data is normally distributed means all sorts of nifty statistical tests can be invented.  Even better, the tests work pretty well even if the distributions are only approximately normally distributed (meaning sort of close to normal but maybe off a little).

"60% of children learn to read by age 7" might be a result of a statistical test...

Source: http://davidmlane.com/hyperstat/A25329.html, retrieved July 12, 2010

Back to grading on the curve

Below is an example of a bell shaped curve.

Let's say a teacher gives a test to a class of students and the mean (or average) score is 80 and the standard deviation is 5

According to the empirical rule, 68% of the students should fall within ± one standard deviation of the mean.  If you look at the curve, those students will get C's (C is meant to show "average performance".

Move out to two standard deviations away from the mean, and you have the B's to the right and the D's to the left.  Move out to three standard deviations from the mean and you have the A's to the right and the F's to the left.

You can set it up as a grading scale:

100 -  91  = A

90  -  85  = B

84  -  76  = C

75  -  70  = D

69  -  00  = F

The key to grading on the curve means you will always have at least one A and one F and the majority of a class will be C's

And the issue of someone "blowing the curve"?  It means a student gets such a high score that the rest of the curve will be skewed to the right, meaning the range for C's and D's will be much higher than the usual 60 - 70%.

Are you getting irritated?

Say you are in a class of 17 students and the mean and standard deviation follow the curve above.

17 x 68% = 11.56 or 12 students will get a C and B's, D's, A's and F's will be spread among 5 students.  Depending on the instructor's preferences, usually it works out to one F, one A, two B's and one D.  Not much chance of improving your gradepoint average!

Examples from the real world

Grading is one example - who uses the bell curve other than sadistic professors?

Just to name a few:

• Feeding milk cows on dairy farms - weights are calculated and rations are regulated based on where Bessie falls on the distribution
• Modern portfolio theory uses the bell shaped curve (or normal distribution to assess an individual stock's performance relative to a group of stocks
• Human resources departments may base decisions on the assumption that factors such as employee job satisfaction, turnover rates and absentee rates are normally distributed

BIG Z-Scores or a way to deal with those curve-busters...

How do you measure an unusual value in your data set?

Statistics geeks call an unusual value an "outlier".

According to Wikipedia, an outlier in statistics is a data point numerically distant from the rest of your data - a rogue point way outside the norm.

You should always be on the lookout for outliers in your data sets.  Outliers can seriously skew your curves to the left or to the right.

Say you are looking at a group of men's weights and part of your sample includes a Sumo wrestler:

Might skew your results a bit?

Malcolm Gladwell wrote an entire book about outliers and their incredible influence on modern culture - an interesting read...(I include the link to his website in case you are interested!)

So - how do you measure an outlier?

A z-score gives us an indication of how unusual a value is because it tells us how far a value sits from the mean.

A z-score of 1 tells us the data value is 1 standard deviation above the mean.  So the bigger the z-score, the more unusual the data point (negative or positive)

Computing a z-score is demonstrated in the following video:

Source: http://en.wikipedia.org/wiki/Outlier, retrieved July 16, 2010

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