The Normal Distributions Chapter 2: The Normal Distributions

Red shaded region represents an approximation of the fraction of scores between 6 and 8 6>scores>8 same as 6<scores>8 Distribution of data can be approximated by a smooth density curve

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

Median

Symmetric density curve. Area0.25 Area0.25 Area0.25 Area0.25 Q1 Q2

Density curve for uniform distribution 1/5 or 0.2 because total area must equal 1 If d = 7 and c =2, what would be the height of the curve? Area = l x w = (7-2) x 1/5 = 1

Normal Distributions All Normal distributions have this general shape. m indicates the mean of a density curve. s indicates the standard deviation of a density curve.

Three normal distributions Differing m results in center of graph at different location on the x axis Differing s results in varying spread Change to page 274

Inflection points -1s +1s Mean

(aka – The Empirical rule) Normal distributions are abbreviated as ; N(m,s) The Normal distribution with mean of 0 and standard deviation of 1 is called the standard Normal curve; N(0,1)

Example What is the z score for Iowa test score of 3.74? N(6.84,1.55) z = (x-m)/s 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 Change to page 285

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% Change to page 291

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 = m + zs

What score is the 90th percentile for N(504,22)? Example What score is the 90th percentile for N(504,22)? X = zs + m = 1.28(22) + 504 = 532.16 m z

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   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