Section 1.4The Normal Distributions

Topics

Density curves Normal distributions The 68-95-99.7 rule The standard normal distribution Normal distribution calculations Standardizing observations Normal quantile plots

Density curves

Density curve

A density curve is a curve that:

is always on or above the horizontal axis has an area of exactly 1 underneath it

A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values on the horizontal axis is the proportion of all observations that fall in that range.

Density curves

Definitions Mode – Location where the curve is high-peaked Median – the equal-areas point. Half the area on

each side. Mean – the balance point of the density curve

◦ Think of placing a wedge so that the density would balance like on a see-saw or teeter totter

◦ Hard to visually find the mean for skewed curves

Mathematical formulas are used to calculate the mean, median, standard deviation, etc.

Normal Density Curve

A right skewed density curve

Mean is the balance of the density curve

Symbols

μ – mean of the idealized distribution (of the density curve)

σ – standard deviation of the idealized distribution

– mean of the actual observations (sample mean)

s – standard deviation of the actual observations (sample standard deviation)

x

Normal Distribution

Symmetric, unimodal, bell-shaped Characterized by mean (μ) and stdev (σ). Mean is the point of symmetry Can visually speculate σ (inflection point?)

Good description of many real variables (test scores, crop yields, height)

Approximates many other distributions

Normal Distributions

Formula (Redundant in this class)

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21( )

2

x

f x e

Normal Distribution

Described only by mean and standard deviation Instead of writing it out each time, we shorthand

Normal using N and put the mean and stdev in parenthesis:

general normal: N(µ, σ)standard normal: N(0,1)

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In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ.

Approximately 95% of the observations fall within 2σ of µ.

Approximately 99.7% of the observations fall within 3σ of µ.

The 68-95-99.7 Rule

Example: heights of young women

The distribution of heights of young women aged 18 to 24 is approximately normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.

Between what two points do 68% of the women fall into? 95%? 99.7%?

Finding probabilities for normal data

Tables for normal distribution with mean µ = 0 and stdev σ = 1 (N(0,1)) are available◦ see page T-2 near the back of the book

1. First learn how to find out different types of probabilities for N(0,1) (standard normal curve).

2. Then learn to convert ANY normal distribution to a standard normal and find the corresponding probability

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Z .00 .01 .02

0.7 .7580 .7611 .7642

0.8 .7881 .7910 .7939

0.9 .8159 .8186 .8212

P(z < 0.81) = .7910

• Table always give the area to the left• Suppose we want to find the proportion of observations from the

standard Normal distribution that are less than 0.81.

The Standard Normal Table

Examples

What proportion of observations on a standard normal variable Z take values

◦ less than 2.2 ?

◦greater than -2.05 ?

What about backwards?

If I give you a probability, can you find the corresponding z value?

called percentiles

◦ What is the z-score for the 25th percentile of the N(0,1) curve?

◦ 90th percentile?

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All Normal distributions are the same if we measure in units of size σ from the mean µ as center.

If a variable x has a distribution with mean µ and standard deviation σ, then the standardized value of x, or its z-score, is

μx

z-

The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. That is, the standard Normal distribution is N(0,1).

Standardizing Observations

Standardizing

Example: Compute the standardized scores for women 70 inches tall and 60 inches tall. (μ=64.5 σ=2.5)

Example

For SAT scores on individual sections, the scores are approximately normal with mean 500 and standard deviation 100. ◦ What percent of students score a 700 or higher?◦ What percent of students score in between 600 and

700?◦ How high must you score on a section to be in the top

1% of all test takers?◦ Mark’s score on one section was the 68th percentile,

what score did he get?

One way to assess if a distribution is indeed approximately Normal is to

plot the data on a Normal quantile plot or QQplot from SAS.

The data points are ranked and the percentile ranks are converted to z-

scores with Table A. The z-scores are then used for the x-axis against

which the data are plotted on the y-axis of the Normal quantile plot.

If the distribution is indeed Normal, the plot will show a straight

line along the 45 degree line, indicating a good match between

the data and a Normal distribution.

Systematic deviations from a straight line indicate a non-Normal

distribution. Outliers appear as points that are far away from the

overall pattern of the plot.

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Normal Quantile Plots

Summary (1.3)

Hypothesized mathematical models for distributions: Density curves

Normal Distribution Evaluating probabilities for standard normal

distribution Evaluating probabilities for ANY normal

distribution by converting it to a standard normal distribution

Normal quantile plot (qqplot)