Chapter 3

The Normal Distributions

Chapter 3 Objectives

• Be able to approximately locate mean and median on a density curve

• Recognize the Normal distribution, esimate mean and SD by eye

• Use the 68-95-99.7 Rule• Find a z-score and interpret it• Given mean and SD, calculate the proportion

above or below a z-score, or the proportion between 2 z-scores

• Given a proportion, be able to calculate the data point with that proportion above or below it.

Describing a Distribution

We now have a clear strategy for exploring data on a single quantitative variable:

1. Always plot your data - make a graph.

2. Look for the overall pattern and for striking deviations (outliers, gaps.)

3. Calculate an appropriate numerical summary to briefly describe center and spread

Rulers and measurement

• We’re going to be spending more time with standard deviation, as a ruler.

• We’re going to use it as a unit of measure, just like inches, cubits, and so on.

This kindNot this kind

Mystery scores…• Test 1: Class Mean: 75, Class SD: 7

You scored 2 standard deviations above the mean.What’s your score?

• Test 2: Class Mean:75, Class SD: 7You scored a 70.

How many SD above/below the mean are you?

How should you calculate this?

Z-scores

• A Z-score tells you how many standard deviations above or below a mean a data value is.

• Remember it this way:

• It allows us to compare data values, even ones from different data sets.

Important Formula!!

zdatavalue mean

SD

Who says you can’t compare Apples and Oranges??

• Actually, we can.• All it takes is a standard deviation.• Example: You have an apple and an orange.

They each weigh 12 ounces.– Which one is bigger?– We really mean: which one is comparatively bigger?– What if I tell you:

• The apple is 2 standard deviations above the mean apple weight

• The orange is 1 standard deviation above the mean orange weight.

Placement Exams

• What we’re really doing is standardizing variations.

An incoming freshman took her college’s placement exams in French and mathematics.

She scored:French: 82 Overall mean: 72, SD: 8Math: 86 Overall mean: 68, SD: 12

On which exam did she do better compared with other freshmen? Explain!

Standardizing Variations

• Sports: – Joe runs the 100 meter dash 2 seconds

faster than average for his school. (mean: 12s, SD: 1s)

– Jane jumps 3 feet further in the long jump than average for her school. (mean: 24ft, SD: 2ft)

– Who’s better at their sport?– We need to compare them to the mean for

their sport, and then measure how far away from the mean they are, in terms of the spread of the distribution for their sport.

Hmm…Interesting…

• What makes a z-score interesting? – Points far away from the rest of the data are

generally more interesting.– Is a z-score of 1 interesting?

• Standard deviation is usually > IQR for symmetric, so more than 50% of data is within 1 SD of mean.

– What about a z-score of 3?• That’s pretty far out! (remember- 1.5IQR was an

outlier.

– How often do we expect to see big z-scores?• To answer this, need to model the data

distribution.

What’s a density curve?

• A: A MODEL that describes a distribution.– Gives the overall pattern– Area under curve = 1.0– Lots of options for shape

The Normal Model• You’ve heard of the “bell curve”

– IQ, grading on a curve– In Statistics, called the Normal Model.

• What is it?– The Normal Model is an idealized description,

a model, of a distribution that is:• Symmetric• Unimodal• Bell-shaped

• Warning: Many sets of data follow a normal model, but many do not.

Not Everyone’s Normal, but a lot are…

• The Normal Model is actually a good description for things like:– SAT scores– Psych tests– IQ scores– Things in biology, like height and weight– Chance outcomes

• Works well to model roughly symmetric, unimodal distributions.– Need to meet Nearly Normal Condition to use Normal

Model!!

Remember it’s a Model• We use special notation for the model:

– tells us the center of the model, the mean.– tells us the standard deviation of the model

• Convention: We’ll use Greek letters for models. These numbers are NOT calculated from data.– Parameters

• The mean is located at the center of the symmetric curve and is the same (approximately) as the median.

• Changing without changing moves the normal curve along the horizontal axis without changing its spread.

N

So what’s the Normal Model good for?

• In a Normal Model, the area under the curve over an interval represents the proportion of observations in that interval.

• We can find how much of the data we expect to be – Above a given value/z-score– Below a given value– Between 2 given values

• Example: Women’s heights are N(64.5”,2.5”)– What proportion of women should be taller than 64.5”?

64.5 6762

The Standard Normal Model and Another z-score formula…

• So you might ask, how do we find that area??– Integrate! Calculus!– Just kidding. But you could.

• Every Normal Model is different- centered at a different with a different . – Different centers, different spreads

• If we calculate the z-score for all of our data points, we standardize the curve. Do the same thing with the Model and we get:

2

2

1

2

1)(

x

exf

Standard Normal Model

• Because all Normal distributions share the same properties, we can STANDARDIZE our data to turn any Normal Curve in the standard Normal Curve, N(0,1)

• N(0,1)

• Inflection point – distance of 1

zy

Same formula,new form

68-95-99.7 Rule• There is a nice approximation for the proportion

of values under the Normal Model. You saw this in action in the activity. This works for any Normal model, not just the Standard Normal Model.

• Approximately 68% of the observations fall within 1 of

• What proportion is between and +1?• What proportion is outside 1of ?

68-95-99.7 Rule

• What percent is outside 3 of ?• What percent is between z=-1 and z=2?• But remember, this is just an approximation!

– Only useful for z = 1,2,3 (positive or negative)

Trees

• A forester measured 27 trees, finding a mean of 10.4 inches and a SD of 4.7 inches. The trees provide an accurate description for the forest, Normal model.– What size are the central 95%?– What percent are < 1 inch? – What percent are between 5.7 and 10.4

inches?

But Be Exact

• Don’t use the 68-95-99.7 Rule except for problems with z=1,2,3, where exact answers not needed.

• You CAN use: Z-table, Normal Curve Applet, JMP

• For all of these problems, draw a picture!• What percent of a standard Normal model is found:

– z > -2.05– z < -0.33– 1.2 < z < 1.8– | z | < 1.28

IQs

• Based on the Normal model N(100,16) describing IQ scores, what percent of people’s IQs do you expect to be – Over 80?– Under 90?– Between 112 and 132?

What are the effects of better maternal care on gestation time and premies?

The goal is to obtain pregnancies of 240 days (8 months) or longer.

Example: Gestation time in malnourished mothers

What improvement did we get

by adding better food?

Reversing the procedure

• In a standard Normal model, what value(s) of z cut(s) off the region described?– The lowest 12%– The highest 30%– The highest 7%– The middle 50%

• Tip: When using the table, remember to use the area to the left of the z you’re looking for.

Body temps

• Most people think that the normal adult temp is 98.6. But in 1992, a more accurate figure was reported to be 98.2, with a SD of 0.7. – What fraction of people should be expected to

have body temps above 98.6?– Below what body temp are the coolest 20% of all

people?

mean µ = 64.5"

standard deviation = 2.5" proportion = area under curve=0.25

Example: Women’s heightsWomen’s heights follow the N(64.5″,2.5″)

distribution. What is the 25th percentile for

women’s heights?