CHAPTER 11

• The chapter covers Normal distributions, the 68-95-99.7 rule, standard Normal distribution, using the standard Normal table, inverse Normal calculations, and Normal quantile plots.

Normal Distributions

• Normal or Gaussian distributions are symmetrical, bell-shaped density curves.
• They are defined by a mean μ (mu) and a standard deviation σ (sigma), denoted as N(μ,σ).
• Normal curves are used to model many biological variables.
• The curves can describe a population distribution or a probability distribution.

Family of Density Curves

• When means are the same, the standard deviations differ.
• When the means are different, the standard deviations are the same.

Normal Model Examples

• Human heights, by gender, can be modeled by a Normal Distribution.
• Guinea pig survival times after inoculation of a pathogen does not make a good candidate for a normal model.

The 68-95-99.7 Rule

• All normal curves N(μ,σ) share the same properties.
• Approximately 68% of all observations are within 1 standard deviation (σ) of the mean (μ).
• About 95% of all observations are within 2 σ of the mean μ.
• Almost all, 99.7%, of observations are within 3 σ of the mean.
• Use technology or Table B to work out any other area under a Normal curve.

Examples of the Normal Distribution in Bone Density

• Normal bone density is within 1 standard deviation (z > -1) of the young adult mean or above.
• Low bone mass is when the bone density is 1 to 2.5 standard deviations below the young adult mean (z between -2.5 and -1).
• Osteoporosis is when the bone density is 2.5 standard deviations or more below the young adult mean (z ≤ – 2.5).
• Approximately 16% of young adults have bone densities of -1 or less.
• 16% represents the area to the left of the middle 68% between -1 and +1.

Standard Normal Distribution

• Data can be standardized by computing a z-score: z = (x-μ)/σ.
• If x has the N(μ, σ) distribution, then z has the N(0, 1) distribution.
• A z-score measures the number of standard deviations a data value x is from the mean μ.
• When x is larger than the mean, z is positive.
• When x is smaller than the mean, z is negative.

Standardizing Z-Scores Example

• For Women's heights following the N(64.5",2.5") distribution
• The percentage of women shorter than 67 inches tall (5'6") can be found as follows:
• The mean μ = 64.5", the standard deviation σ = 2.5", and the height x is 67".
• The standardized value of x is z = (67 - 64.5) / 2.5 = 1, which equals 1 standard deviation from the mean.
• Approximately 84% of women are estimated to be shorter than 67".
• This is based on the 68–95–99.7 rule, where 0.68 + half of (1 – 0.68) = 0.84.

Using the Standard Normal Table

• The Standard Normal Table gives the area under the standard Normal curve to the left of any z-value.
• For z = 1.00, the area under the curve to the left of z is 0.8413 which means 84.13% of women are shorter than 67".
• Therefore, 15.87% of women are taller than 67" (5'6").
• Due to curve symmetry, there are two ways to find the area under N(0,1) curve to the right of a z-value.

Using the Standard Normal Table to Find a Middle Area

• Calculate the area between two z-values by finding area under N(0,1) to the left for each z-value from the table
• Afterwards, subtract the smaller area from the larger area.
• The area under N(0,1) for a single value of z is zero.

Blood Cholesterol Example

• Blood cholesterol levels of men age 55 to 64 is approximately Normal with mean 222 mg/dL and standard deviation 37 mg/dL.
• Percent of middle-age men who have high cholesterol (greater than 240 mg/dL) and percent who have elevated cholesterol (between 200 and 240 mg/dL) can be found under the standard Normal curve, the difference gives the answer.

Inverse Normal Calculations

• Inverse Normal calculations can be used to seek the range of values that correspond to a given proportion or area, under the curve.
• To perform them, use technology or the standard Normal table backward
• First find the desired area/proportion in the body of the table.
• Then read the corresponding z-value from the left column and top row.

Inverse Normal Calculation Example

• Hatching weights of commercial chickens can be modeled accurately using a Normal distribution with mean µ = 45 g and standard deviation σ = 4 g.
• To find the third quartile of the distribution of hatching weights:
  • We know μ, σ, and the area under the curve; we want x.
  • Table B gives the area left of z look for the lower 25%. We find z≈ 0.67.
  • Substituting the known values x = 45 + (0.67 * 4) = 47.68
  • Therefore Q3 ≈ 47.7g

Normal Quantile Plots

• To assess if a data set has an approximately Normal distribution, plot the data on a Normal quantile plot (Q-Q Plot).
• The data points are ranked and the percentile ranks are then converted to z-scores.
• The z-scores are then used for the horizontal axis and the actual data values are used for the vertical axis.
• Technology is needed to obtain Normal quantile plots.
• The Normal quantile plot will have roughly a straight-line pattern if the data have approximately a Normal distribution.

This lesson covers normal distributions, including the 68-95-99.7 rule and standard normal distribution. It also explains how to use the standard normal table and perform inverse normal calculations. Normal quantile plots are discussed.