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The Normal DistributionPowerPoint Presentation

The Normal Distribution

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History

- Abraham de Moivre (1733) – consultant to gamblers
- Pierre Simon Laplace – mathematician, astronomer, philosopher, determinist.
- Carl Friedrich Gauss – mathematician and astronomer.
- AdolpheQuetelet -- mathematician, astronomer, “social physics.”

Importance

- Many variables are distributed approximately as the bell-shaped normal curve
- The mathematics of the normal curve are well known and relatively simple.
- Many statistical procedures assume that the scores came from a normally distributed population.

- Distributions of sums and means approach normality as sample size increases.
- Many other probability distributions are closely related to the normal curve.

Using the Normal Curve size increases.

- From its PDF (probability density function) we use integral calculus to find the probability that a randomly selected score would fall between value a and value b.
- This is equivalent to finding what proportion of the total area under the curve falls between a and b.

The PDF size increases.

- F(Y) is the probability density, aka the height of the curve at value Y.
- There are only two parameters, the mean and the variance.
- Normal distributions differ from one another only with respect to their mean and variance.

Avoiding the Calculus size increases.

- Use the normal curve table in our text.
- Use SPSS or another stats package.
- Use an Internet resource.

IQ = 85, PR = ? size increases.

- z = (85 - 100)/15 = -1.
- What percentage of scores in a normal distribution are less than minus 1?
- Half of the scores are less than 0, so you know right off that the answer is less than 50%.
- Go to the normal curve table.

Normal Curve Table size increases.

- For each z score, there are three values
- Proportion from score to mean
- Proportion from score to closer tail
- Proportion from score to more distant tail

Locate the | size increases.z| in the Table

- 34.13% of the scores fall between the mean and minus one.
- 84.13% are greater than minus one.
- 15.87% are less than minus one

IQ size increases.=115, PR = ?

- z = (115 – 100)/15 = 1.
- We are above the mean so the answer must be greater than 50%.
- The answer is 84.13% .

85 < IQ < 115 size increases.

- What percentage of IQ scores fall between 85 (z = -1) and 115 (z = 1)?
- 34.13% are between mean and -1.
- 34.13% are between mean and 1.
- 68.26% are between -1 and 1.

115 < IQ < 130 size increases.

- What percentage of IQ scores fall between 115 (z = 1) and 130 (z = 2)?
- 84.13% fall below 1.
- 97.72% fall below 2.
- 97.72 – 84.13 = 13.59%

The Lowest 10% size increases.

- What score marks off the lowest 10% of IQ scores ?
- z = 1.28
- IQ = 100 – 1.28(15) = 80.8

The Middle 50% size increases.

- What scores mark off the middle 50% of IQ scores?
- -.67 < z < .67;
- 100 - .67(15) = 90
- 100 + .67(15) = 110

Memorize These Benchmarks size increases.

The Normal Distribution size increases.

The Bivariate Normal Distribution size increases.

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