The Normal Distributions Section 1.4 The 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)

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)

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)

The 68-95-99.7 Rule 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.

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 First learn how to find out different types of probabilities for N(0,1) (standard normal curve). Then learn to convert ANY normal distribution to a standard normal and find the corresponding probability

The Standard Normal Table 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. P(z < 0.81) = .7910 Z .00 .01 .02 0.7 .7580 .7611 .7642 0.8 .7881 .7910 .7939 0.9 .8159 .8186 .8212

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?

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

Normal Quantile Plots 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.

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)