Section 6.1

Introduction to the Normal Distribution

Entirely rewritten by D.R.S., University of Cordele.

All About the Normal Distribution

It is Bell Shaped

All About the Normal Distribution

It is Symmetric – the Left Side and the Right Side are mirror images of each other with respect to the vertical line at the peak, in the middle.

Fold it over at the center line and the curvy parts will match exactly.

All About the Normal Distribution

It goes on FOREVER, with the horizontal axis as an ASYMPTOTE (forever approaching but never touching nor crossing the axis)

The curve gets closer and closer and closer and closer to the horizontal axis but they never touch; they never cross.

The curve gets closer and closer and closer and closer to the horizontal axis but they never touch; they never cross.

All About the Normal Distribution

The total area under the entire curve – even counting the stretch to ∞ and to – ∞, is EXACTLY 1.0000000000, the same as a 1-by-1 square!!!

TotalArea

= 1.00000

exactly !!!

width= 1

length = 1

Area of thissquare =

1 x 1 = 1.00000

exactly !!!!

and this just happens to tie in with the important fact about probability distributions: that the sum of the probabilities in the probability column must equal exactly precisely 1.00000 !!!!

The z Axis

The z-axis always has 0 in the middle. If you make it go from -3 to +3 with steps of size 1, it fits most problems just fine.

z -3 -2 -1 0 1 2 3

It is unusual to see a z value beyond –3 or +3 but it happens.

Beyond –4 and +4 is extremely rare! Be suspicious if it happens during your work. It’s possible but extreme.

The x Axis

The x-axis lines up with the z-axis but it has different numbers because there are many different normal distributions. The numbers depend on the mean and on the standard deviation.

x

z -3 -2 -1 0 1 2 3

The x Axis

Suppose it’s the mean test score of 75 and a standard deviation of 8. 75 is in the middle…

x 75

z -3 -2 -1 0 1 2 3

The x Axis

Suppose it’s the mean test score of 75 and a standard deviation of 8. 75 is in the middle; each step up is +8

x 75 83 91 99

z -3 -2 -1 0 1 2 3

The x Axis

Suppose it’s the mean test score of 75 and a standard deviation of 8. 75 is in the middle; each step up is +8 and each step down is -8 fromthe mean

x 51 59 67 75 83 91 99

z -3 -2 -1 0 1 2 3

The x Axis

Similarly, if it’s mean life span of 81.4 years with a standard deviation of 4.3 years, we have this x-axis:

x 68.5 72.8 77.1 81.4 85.7 90.0 94.3

z -3 -2 -1 0 1 2 3

Formula to convert value to value

Suppose the mean and the standard deviation . What’s the score that corresponds to an value of 77.1?

x 68.5 72.8 77.1 81.4 85.7 90.0 94.3

z -3 -2 -1 0 1 2 3

Formula:

Formula to convert value to value

Suppose the mean and the standard deviation . What’s the score that corresponds to an value of 85.0?

x 68.5 72.8 77.1 81.4 85.7 90.0 94.3

z -3 -2 -1 0 1 2 3

Formula:

Formula to convert value to value

Suppose the mean and the standard deviation . What’s the score that corresponds to an value of 2.00?

x 68.5 72.8 77.1 81.4 85.7 90.0 94.3

z -3 -2 -1 0 1 2 3

Formula:

Going the other

way: z to x…

Formula to convert value to value

Suppose the mean and the standard deviation . What’s the score that corresponds to an value of -1.37?

x 68.5 72.8 77.1 81.4 85.7 90.0 94.3

z -3 -2 -1 0 1 2 3

Formula:

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Score: is how many standard deviations away from the mean?

If you know the x value• Population:

• Sample

To work backward from z to x• Population

• Sample

These formulas agree with the labeling of the axes you did in the Empirical Rule and Chebyshev’s Theorem problems. In those problems, the z values were always nice integers: -3, -2, -1, 0, 1, 2, 3.

score values

Typically round to two decimal places.– Don’t say “0.2589”, say “0.26”

If not two decimal places, pad– Don’t say “2”, say “2.00”– Don’t say “-1.1”, say “-1.10”

scores are almost always in the interval . Be very suspicious if you calculate a score that’s not a small number.

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How are the formulas related?

.1) Start with this definition.

2) Multiply each side by .

3) Add to each side.

4) Did you arrive at ?

Input x, output z.

Input z, output x.

Standard Score answers the question“How does my compare to the mean?”

“Am I in the middle of the pack?”“Am I above or below the middle?”“Am I extremely high or extremely low?” Score is the measuring stickIf z= 0, then I’m ________________________.If z > 0,then I’m ________________________.If z < 0, then I’m ________________________.z is almost always between _____ and _____.

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scores caution with negatives

Example: compare test scores on two different tests to ascertain “Which score was the more outstanding of the two?”Be careful if the scores turn out to be negative. Which is the better performance? or ?Stop and think back to your basic number line and the meaning of “<“ and “>”

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Excel STANDARDIZE function to convert a data value (x) to a standard score (z)

The value is sometimes called

“The Standard Score”.

So “standard-ize” takes an value and “standard-izes” it by

changing it to a score.