Chapter 2:The Normal Distributions

Distribution of data can be approximated by a smooth density curve

Red shaded region represents an approximation of the fraction of scores between 6 and 8

6>scores>8 same as

6<scores>8

Area A, represents the proportion observations falling between values a and b

Median

Area0.25

Area0.25

Area0.25

Area0.25

Q1 Q2

Symmetric density curve.

Density curve for uniform distribution

If d = 7 and c =2, what would be the height of the curve?

1/5 or 0.2 because total area must equal 1

Area = l x w = (7-2) x 1/5 = 1

Normal Distributions

All Normal distributions have this general shape.

indicates the mean of a density curve.

indicates the standard deviation of a density curve.

Three normal distributions

Differing results in center of graph at different location on the x axis

Differing results in varying spread

Mean-1 +1

Inflection points

Normal distributions are abbreviated as ; N(

The Normal distribution with mean of 0 and standard deviation of 1 is called the standard Normal curve; N(0,1)

(aka – The Empirical rule)

Example

N(6.84,1.55)

What is the z score for Iowa test score of 3.74?z = (x-

z = (3.74 – 6.84)/1.55

z = -2

This says that a score of 3.74 is 2 standard deviation below the mean

Using the 68 – 95 – 99.7 rule to solve problems

Example

What percentage of scores are greater than 5.29?

Finding Normal Percentiles by• Table A is the standard Normal table. We have to

convert our data to z-scores before using the table.• The figure shows us how to find the area to the left

when we have a z-score of 1.80:

Using Table A to find the area under the standard normal curve that lies (a) to the left of a specified z-score, (b) to the right of a specified z-score, and (c) between two specified z-scores

0228.9772.1

)00.2(1

)00.2(1)00.2(

50

30003100)3100(

zP

zPzP

zPxP

• Say a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3,000 and = 50)

• What is the probability that the car will go more than 3,100 yards without recharging?

Determine the percentage of people having IQs between 115 and 140

P[115< x < 140]

P[(115-100)/16 < z < (140-100)/16]

P[0.94< z < 2.50]

= 0.9938 – 0.8264

= 0.1674 = 16.74%

From Percentiles to Scores: z in Reverse

• Sometimes we start with areas and need to find the corresponding z-score or even the original data value.

• Example: What z-score represents the first quartile in a Normal model?

From Percentiles to Scores: z in Reverse

• Look in Table A for an area of 0.2500.• The exact area is not there, but 0.2514 is pretty close.

• This figure is associated with z = –0.67, so the first quartile is 0.67 standard deviations below the mean.

• To unstandardize; solve x = + z

Example

What score is the 90th percentile for N(504,22)?

z

X = z +

= 1.28(22) + 504

= 532.16

Methods for Assessing Normality

• If the data are normal– A histogram or stem-and-leaf display will look like the normal

curve– The mean ± s, 2s and 3s will approximate the empirical rule

percentages. (68%,95%,99.7%)– The ratio of the interquartile range to the standard deviation

will be about 1.3– A normal probability plot , a scatterplot with the ranked data on

one axis and the expected z-scores from a standard normal distribution on the other axis, will produce close to a straight line

Errors per MLB team in 2003– Mean: 106

– Standard Deviation: 17

– IQR: 22

29.117

22

s

IQR

15755

511063

14072

341062

12389

17106

sx

sx

sx

22 out of 30: 73%

28 out of 30: 93%

30 out of 30: 100%

A normal probability plot is a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis

• A skewed distribution might have a histogram and Normal probability plot like this:

Khan Academy Videos

http://www.khanacademy.org/math/statistics/v/introduction-to-the-normal-distribution http://www.khanacademy.org/math/statistics/v/ck12-org-normal-distribution-problems--qualitative-sense-of-normal-distributions

http://www.khanacademy.org/math/statistics/v/ck12-org-normal-distribution-problems--z-score

http://www.khanacademy.org/math/statistics/v/ck12-org-normal-distribution-problems--empirical-rule

http://www.khanacademy.org/math/statistics/v/ck12-org-exercise--standard-normal-distribution-and-the-empirical-rule

http://www.khanacademy.org/math/statistics/v/ck12-org--more-empirical-rule-and-z-score-practice