13 1normal

The Normal Distribution

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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

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Let X be a continuous random variable that follows a normal distribution
with

  • mean = $\mu$
  • standard deviation = $\sigma$

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Notation:

… … $X \sim N \big(\mu, \; \sigma^2 \big)$

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WARNING!!

Most of the calculations will involve the standard deviation, but the notation shows the variance!

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Properties of the Normal Distribution

The equation for the normal distribution is:

13.1eqn1.gif

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13.1grph1.gif

Notice that the distribution has the following properties:

  • Symmetrical
  • mean = $\mu$
  • median = $\mu$
  • mode = $\mu$
  • x-axis is an asymptote
  • Maximum value is at $\left( \mu, \; \dfrac{1}{\sigma \sqrt{2\pi}} \right)$
  • $Pr(a < X < b) = \displaystyle{ \int_a^b \; f(x) \; dx}$
  • $\displaystyle{ \int_{-\infty}^{+\infty} \; f(x) \; dx} = 1$

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Notice the curve is very close to the x-axis at $\mu \pm 3\sigma$
… but not touching because the x-axis is an asymptote

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NOTE:

  • $\mu$ represents a translation of the curve to the right $\big( \text{if } \mu > 0 \big)$
  • $\sigma$ represents a dilation of the curve from the y-axis (in the x-direction)
  • and $\sigma$ also represents a dilation of $\dfrac{1}{\sigma}$ from the x-axis (in the y-direction)

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Standard Normal Distribution

When $\mu = 0 \text{ and } \sigma = 1$ we get the Standard Normal Distribution

We use Z for the Standard Normal Distribution

… … … $Z \sim N \big(0,\; 1\big)$
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Therefore the equation of a Standard Normal Distribution is:

13.1sndeqn.JPG

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13.1nrmlgrph.JPG

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Note: the Standard Normal Distribution is

  • centred on the y-axis $\big( \mu = 0 \big)$
  • the x-axis is an asymptote $\big( y = 0 \big)$
  • is close to the x-axis at $x = \pm3$
  • Has a mean, median, mode at $x = 0$

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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).

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68% Confidence Interval

Approximately 68% of the data lies within one standard deviation of the mean.

… … $Pr \big( \mu - \sigma < X < \mu + \sigma \big) \approx 0.68$

13.1conf1.gif

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95% Confidence Interval

Approximately 95% of the data lies within two standard deviations of the mean.

… … $Pr \big( \mu - 2\sigma < X < \mu + 2\sigma \big) \approx 0.95$

13.1conf2.gif

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99.7% Confidence Interval

Approximately 99.7% of the data lies within three standard deviations of the mean.

… … $Pr \big( \mu - 3\sigma < X < \mu + 3\sigma \big) \approx 0.997$

13.1conf3.gif

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Example 1

Scores from a maths test are normally distributed with

a mean of 84 and a standard deviation of 4

… … … $X \sim N\big(84, \; 16\big)$

Therefore its equation is this:

13.1eg1eqn.JPG

a) .. Sketch the graph of X

b) .. Find the approximate percentage of scores between 80 and 88

c) .. Find the approximate percentage of scores below 84

d) .. Find the approximate percentage of scores above 92

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Solution

a) .. Sketch the graph of X

… … … Use $\mu \text{ and } \sigma$ to produce the scale

… … … A vertical line at \mu can help to keep the graph symmetrical

… … … Don't forget to label the turning point

… … … The bell shape should approach the axis close to $\mu \pm 3\sigma$.

13.1eg1grph.gif

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b) .. Find the approximate percentage of scores between 80 and 88

… … … 80 is 4 below the mean $\big( \mu - \sigma \big)$

… … … 88 is 4 above the mean $\big( \mu + \sigma \big)$

… … … so

… … … $80 < x < 88$ is one standard deviation each side of the mean

… … … so

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

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c) .. Find the approximate percentage of scores below 84

… … … median = 84

… … … so

… … … 50% of the scores are below 84

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d) .. Find the approximate percentage of scores above 92

… … … 92 is 8 above the mean \big( \mu + 2\sigma \big)

… … … 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}

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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}
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To calculate: $Pr \big( \text{lower } < X < \text{ upper} \big)$

… … normalcdf (lower, upper, $\sigma$, $\mu$ )

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Caution

Do not write calculator notation as a part of your answer for SACs or Exams.

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Example 1 (continued)

For the variable: $X \sim N \big(84,\; 16\big)$

Use the CAS to find these percentages correct to one decimal place

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

b) .. Find the % of scores below 84

c) .. Find the % of scores above 92

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Solution

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

… … … enter: $\text{normalcdf lower } = 80, \text{ upper }= 88,\; \sigma = 4,\; \mu = 84$

… … … $Pr \big(80 < X < 88\big) = 0.6827 = 68.3 \%$

13.1eg1calc1.gif

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b) .. Find the % of scores below 84

… … … enter: $\text{normalcdf lr } = -\infty. \text{ ur } = 84,\; \sigma = 4,\; \mu = 84$

… … … $Pr\big( X < 84\big) = 0.5 = 50.0 \%$
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c) .. Find the % of scores above 92

… … … enter: $\text{normalcdf lr } = 92, \text{ ur }= \infty,\; \sigma = 4.\; \mu = 84$

… … … $Pr\big(X > 92\big) = 0.0228 = 2.3\%$

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Note: Do not write calculator notation in your answers in tests/exams. I include it here so you can follow what is going on.

Working and answers should be written in mathematical notation:

eg your answer might look like:

… … $\text{For } X \sim N\big(84, \; 16\big)$

… … $Pr(80 < X < 88) = 0.6827$

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Example 2

Sketch the graph of $X \sim N\big(20, \; 9\big)$

Solution:

… … $\mu = 20$

… … $\sigma^2 = 9 \qquad so \quad \sigma = 3$

… … Mark in 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 $\big(\mu \pm 3\sigma \big)$

… … Maximum is at $\left( \mu, \; \dfrac{1}{\sigma \sqrt{2\pi}} \right)$

13.1eg2grph.gif

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Don't Forget!

… … $Pr(X < a) = Pr(X \leqslant a)$

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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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