Normal Distributions

Section 2.1.2

Starter• A density curve starts at the origin and

follows the line y = 2x. At some point on the line where x = p, the curve drops vertically to return to the x axis.

1. Draw the curve.

2. What is the value of p?

3. What is the median of the curve?– Show it as a vertical line on the curve.

Answer

2

If x = p, then y = 2p

Since area = 1 in a density curve:

11 ( )(2 )

2

1

1

p p

p

p

2

If M is the x value of the median, then 2M is the height of the triangle.

Since one half the area is to the left of the median, then triangle area is one half.

1.5 ( )(2 )

2

.5

1 2.5 .707

2 2

M M

M

M

Today’s Objectives

• Draw a normal curve and show μ, µ±σ, µ±2σ, µ±3σ on the graph

• Use the Empirical Rule (a.k.a. 68-95 Rule) to answer questions about percents and percentiles

Normal Curves

• Draw a bell-shaped curve above an x axis• Draw the vertical line of symmetry

– Label the x axis “μ” at this point

• Show the two “inflection points” on the curve• The left inflection point is where the curve stops getting more

steep and starts getting less steep• The other is symmetric to it about the mean line

– Label the x axis “µ+σ” and “µ-σ” below the points

• Using the same scale, label the x axis with µ plus and minus 2σ and 3σ

Normal Curves• Normal curves are special case density

curves– The area under the curve is 1

• This is true of ALL density curves

– The curve is symmetric and “bell-shaped”• So mean = median• We normally speak of the mean rather than median

– The inflection points of the curve are one standard deviation (σ) above and below the mean

The Empirical Ruleor: 68-95 Rule

• In a normal distribution with mean μ and standard deviation σ:– This is called the N(μ, σ) distribution– About 68% of the observations fall within σ of μ– About 95% fall within 2σ of μ– About 99.7% fall within 3σ of μ

Example

• Suppose the heights of American men are known to be N(69 in, 2.5 in)– Draw the normal curve and label the axis– What percent of men are between 69 inches

and 71.5 inches tall?• Since 68% are between 66.5 and 71.5, and the

graph is symmetric, there are 34% between 69 and 71.5 inches tall.

Example Continued

• What percent of men are taller than 74 in?– Since 95% of observations fall within ±2σ of μ,

then 5% fall outside those borders.– By symmetry, 2.5% fall more than 2σ above

the mean.• In this case, that is 69 + 2 x 2.5 = 74 in• So 2.5% of men are taller than 74 in

Example Concluded

• In what percentile is a man who is 71.5 in tall?

• Recall that “percentile” means the percent of observations equal to or less than the specified value

– By definition, 50% fall below 69 inches – 71.5 inches is one σ above the mean, so 34%

must fall between 69 and 71.5– Thus 71.5 inches is the 84th percentile

• 50 + 34 = 84

Exploring Normal Data

• 50 Fathoms Demo 3– What Do Normal Data Look Like?

Today’s Objectives

• Draw a normal curve and show μ, µ±σ, µ±2σ, µ±3σ on the graph

• Use the Empirical Rule (a.k.a. 68-95 Rule) to answer questions about percents and percentiles

Homework

• Read pages 73 – 77

• Do problems 6 – 9