Normal Distributions

Remember rolling a 6 sided dice and tracking the results This is a uniform distribution (with certain characteristics)

A histogram is used to display a normal distribution

Histograms: The vertical axes of a histogram contains the frequency (number) The horizontal axes of a histogram contains the bins into which each piece of data must fall

Bin width: The width of each interval of the histogram. They should be equal. Try to avoid bins with a frequency of zero Do not “hit the post”

Not a histogram

Who likes popcorn?

Find the sum of three dice in 50 rolls Group the results 3 and 4 in one bin, 5 and 6 in another bin…and so on

Normal Distribution

Normal Distribution Make a note of the characteristics and Diagram that are on page 425.

The Normal Curve Since a Normal Distribution is described in terms of percentages, we define the area under the Normal Curve as 1 (100%) The percentage of data that lies between two values in a normal distribution is equivalent to the area that lies under the normal curve between these two posts.

Z- Scores

Consider the following situation: A school gives a scholarship for the highest mark in Data Management The student must be taking all three maths as well to receive the award (so they may get beat in DM) Caley, who took MDM first semester, received 84%. Lauren, who took MDM second semester, received 79% Who should get the MDM award?

It depends… Both student’s marks must be compared on the same scale. Think Canadian and American money.

Results can be written in terms of “standard deviations away from the mean.” (Z-score) This allows for effective comparisons

Conversion to z-score Z = x - x X: result X: mean : Standard Deviation s s

Caley: 84%, CA: 74%, sd: 8 Z = = 1.25 That means 84% is 1.25 standard deviations above the mean (double check…)

Z = = 1.94 That means 79% is 1.94 standard deviations above the mean (a better relative grade) Lauren: 79%, CA: 60%, sd: 9.8

mean

Suppose you received 75% as a final mark in a class. You want to know what percentage of students were below your grade in your class. Assume your class follows a normal distribution. Example 2

If we assign the area under the standard normal curve to be 1, then the percentage of results less then a given data point, will be equal to the area under the curve to the left of the equivalent z-score post.

The areas under the curve are calculated an summarized on page 606 Convert: x = 75%, CA: 70, SD: 6 Z = 0.83

Look up 0.83 in the chart That means 79.67% of the grade were below your grade of a 75%

Do example 2 and 3 on pg 426 together pg 146 z score info

Notice: Since the area under every normal curve equals 1. The percent of the data that lies between 2 specific values, a and b, is the area under the normal curve between endpoints a and b

a b b z-score area – a z-score area

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