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The idea of a normal distribution is a very important, and amazing, idea in mathematics. Why is it important?
From Regents Test Prep website:
From Regents Test Prep website:
A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, etc. Certain data, when graphed as a histogram (data on the horizontal axis, amount of data on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.
A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, etc. Certain data, when graphed as a histogram (data on the horizontal axis, amount of data on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.
It is hard to believe in some cases, but this normal distribution occurs naturally in so many different situations.
- If you shoot a gun at a target many, many times and measure how far each shot misses the bullseye, your results will form a normal distribution.
- If you set up a robot to shoot a gun using laser sights at a target in a controlled indoor shooting range, tiny variations in the water molecules in the air and the drag as the bullet leaves the barrel and the way the gunpowder ignites in each particular casing will cause slight changes in where each bullet hits. Measure those changes and you will find a normal distribution.
- Throw a hula hoop into a field of flowers and count how many are in the hoop. When you count many, many times, the results will create a normal distribution.
- Catch every deer in Pennsylvania and comb through their fur to identify the number of deer ticks each one carries. While you are at the doctor getting a shot for lyme disease you can compile the results and see a normal distribution.
Normal distributions are hard to accept sometimes. How do the flowers know where to grow? How do all the people in a town coordinate how tall to grow so that their heights form a normal distribution (because they do, in any town you choose)? There are a few important ground rules.
Normal distributions are hard to accept sometimes. How do the flowers know where to grow? How do all the people in a town coordinate how tall to grow so that their heights form a normal distribution (because they do, in any town you choose)? There are a few important ground rules.
You need to collect enough data values. This is an idea referred to as the law of large numbers. It means that the patterns in statistics only prove to be true if you look at a big enough collection of data points. If you throw the hula hoop 10 times, it probably won't work. If you throw it 100 times, it might work. If you have 100 people throw it 100 times every day for 100 (seasonally appropriate) days, it will work.
You need to collect enough data values. This is an idea referred to as the law of large numbers. It means that the patterns in statistics only prove to be true if you look at a big enough collection of data points. If you throw the hula hoop 10 times, it probably won't work. If you throw it 100 times, it might work. If you have 100 people throw it 100 times every day for 100 (seasonally appropriate) days, it will work.
You need to avoid human interference. If you ask people their favorite number, you probably will not get a normal distribution. Cultural traditions, sports players, and other factors will influence the number people choose. Normal distributions rely on natural occurrences. (But, normal distributions still work on people. You just need to measure something that people don't choose, like height or number of eyelashes.)
You need to avoid human interference. If you ask people their favorite number, you probably will not get a normal distribution. Cultural traditions, sports players, and other factors will influence the number people choose. Normal distributions rely on natural occurrences. (But, normal distributions still work on people. You just need to measure something that people don't choose, like height or number of eyelashes.)
So these normal distributions happen everywhere. To really see the effects, we need to look at the graph. Across the horizontal axis are the values that you are measuring - flowers or heights or ticks - and up the vertical axis is the frequency of that value.
Again from Regents Test Prep website:
Normal distributions are symmetrical with a single central peak at the mean (average) of the data. The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean. Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
For example,
- If you measure the height of everyone in a town, the average, or mean, might be 68". Or in a Scandinavian town the mean might be 71". Or in an eastern Asian town the average might be 64".
- In each town, though, when you graph how many people are 1/2" taller than average, 1" taller, 2" taller, etc. you will find the same number of people that much shorter as well.
- You will also find less people the further you go out from the mean. These differences, or deviations, from the mean are part of the normal distribution.
- You can calculate the cutoffs for your town's height so that the people match the standard deviations shown.
So the makers of the IQ tests find the mean for their test. They make that result equal a score of 100. Then they measure the deviation of scores that their test returns. They set those scores so that they match the standard deviations. That way, every IQ test results can be compared using the same scale.
So the makers of the IQ tests find the mean for their test. They make that result equal a score of 100. Then they measure the deviation of scores that their test returns. They set those scores so that they match the standard deviations. That way, every IQ test results can be compared using the same scale.
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