12Enormal


 * The Normal Distribution **

The ** Normal Distribution ** is:
 * A very common probability density function
 * A very realistic model of many observed distributions in real life
 * A curve with a symmetrical, bell-shape

Let X be a continuous random variable that follows a normal distribution with mean = m and standard deviation = s.

... ... ** X ~ N( m, s 2 ) **
 * Notation: **

... ... Most of the calculations will involve the __**standard deviation**__, but the notation shows the __**variance**__!
 * WARNING!! **

Properties of the Normal Distribution

The equation for the normal distribution is: Notice that the distribution has the following properties:
 * Symmetrical
 * m = mean = median = mode
 * x-axis is an asymptote

math \\ . \quad \centerdot \text{ Maximum value is at } \left( \mu, \; \dfrac{1}{\sigma \sqrt{2\pi}} \right) \qquad .\\. \\ . \quad \centerdot \; Pr(a < X < b) = \displaystyle{ \int_a^b \; f(x) \; dx} \\ .\\ . \quad \centerdot \; \displaystyle{ \int_{-\infty}^{+\infty} \; f(x) \; dx} = 1 math

Notice the curve is very close to the x-axis at m± 3 s (but the x-axis is an asymptote)

NOTE:
 * m represents a translation of the curve to the right (if m > 0)
 * s represents a dilation of the curve from the y-axis (in the x-direction)
 * and also a dilation of 1/ s from the x-axis (in the y-direction)


 * Standard Normal Distribution **
 * When m = 0 and s = 1 we get the Standard Normal Distribution (see 13.3)
 * We use Z for the Standard Normal Distribution

... ... ... Z ~ N(0, 1)

Therefore the equation of a Standard Normal Distribution is: ... ... ... Notice that the Standard Normal Distribution is
 * centred on the y-axis
 * uses the x-axis as an asymptote
 * is close to the x-axis at about x = ±3
 * Has a mean, median, mode at x = 0


 * Confidence Intervals **

For the normal distribution, the ** confidence intervals ** (introduced in Ch 10 for discrete variables) are accurate enough for most practical purposes (but are still approximate).

68% Confidence Interval

Approximately 68% of the data lies within __**one**__ standard deviation of the mean. math . \qquad Pr \big( \mu - \sigma < X < \mu + \sigma \big) \approx 0.68 \qquad. math

95% Confidence Interval

Approximately 95% of the data lies within __**two**__ standard deviations of the mean. math . \qquad Pr \big( \mu - 2\sigma < X < \mu + 2\sigma \big) \approx 0.95 \qquad. math

99.7% Confidence Interval

Approximately 99.7% of the data lies within __**three**__ standard deviations of the mean. math . \qquad Pr \big( \mu - 3\sigma < X < \mu + 3\sigma \big) \approx 0.997 \qquad. math


 * Example 1 **

Scores from a maths test are normally distributed with a **mean of 84** and a **standard deviation of 4**

... ... ... X ~ N(84, 16)

Therefore its equation is this: ... ... ...


 * Sketch the graph (don't forget to label the turning point) **
 * Use the mean and standard deviation to produce the scale


 * The bell shape should approach the axis close to 3 standard deviations each side of the mean.


 * Don't forget to label the turning point.


 * a) Find the approximate percentage of scores between 80 and 88 **

... ... 80 is 4 (1 ´ // s // ) below the mean ... ... 88 is 4 (1 ´ // s // ) above the mean so ... ... 80 < x < 88 is one standard deviation each side of the mean

So 68% (approx) of the scores are between 80 and 88


 * b) Find the approximate percentage of scores below 84 **

... ... median = 84

so 50% of the scores are below 84


 * c) Find the approximate percentage of scores above 92 **

... ... 92 is 8 (2 ´ // s // ) above the mean ... ... 92 is two standard deviations above the mean

... ... 95% are within two standard deviations of the mean, so ... ... 5% are outside two standard deviations of the mean

... ... The distribution is symmetrical (half above, half below)

so 2.5% (approx) are above 92

{The other 2.5% are below 76}

Normal Distribution on the Classpad Calculator

From the main screen, go to the ** interactive ** menu and Select ** Distribution ** Select ** Normalcdf {Do NOT use normalpdf} **

Then enter lower limit, upper limit, standard deviation, mean (Using the __**interactive**__ version will make sure you enter values in the correct order)

math \\ . \qquad \textbf{normalcdf} ( \text{lower, upper, } \sigma, \mu ) \\. \\ . \qquad \text{will calculate } Pr( \text{lower } < X < \text{ upper} ) \qquad. math

{On older versions of the Classpad, you will find normalcdf in the Statistics package in the CALC menu (distributions)}

X ~ N(84, 16)
 * Example 1 (continued) **


 * a) Find the % of scores between 80 and 88 **

... ... **enter**: normalcdf lower = 80, upper = 88 sd = 4 ,mean = 84 ... ... Pr(80 < X < 88) = 0.6827 = 68.3%


 * b) Find the % of scores below 84 **

... ... **enter** normalcdf lr = – ¥ ur = 84 sd = 4 mn = 84 ... ... Pr( X < 84) = 0.5 = 50%


 * c) Find the % of scores above 92 **

... ... **enter** normalcdf lr = 92 ur = ¥ sd = 4 mn = 84 ... ... Pr(X > 92) = 0.0228 = 2.3%

Note: Do not write calculator notation in your answers in tests/exams. I include it here so you can follow what is going on. Answers should be written in mathematical notation: eg ... ... For X ~ N(84, 16) ... ... Pr(80 < X < 88) = 0.6827


 * Example 2 **

Sketch the graph of X ~ N(20, 9)

__**Solution:**__

math . \quad \centerdot \text{ Maximum is at } \left( \mu, \; \dfrac{1}{\sigma \sqrt{2\pi}} \right) \qquad. math
 * **Mean** is **20**
 * __Variance is 9__, so **Standard Deviation** is **3**
 * Mark in the scale, using steps of 3 either side of 20
 * Draw the bell-shaped curve, centred at 20
 * The curve should get close to the axis near 11 and 29 ( m± 3 s )

Don't Forget!

math . \qquad Pr(X < a) = Pr(X \leqslant a) \qquad. math

Interactive Approximate Normal Distribution The following link is to an interactive that demonstrates:
 * the difference between an approximate normal distribution and the theoretical normal distribution
 * the way that increasing the sample size makes the approximate distribution approach the theoretical normal
 * the effect of changing the standard deviation on the shape of the curve

Shodr Interactives

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