Section 2.1Section 2.1Density CurvesDensity Curves

Get out a coin and flip it 5 times. Get out a coin and flip it 5 times. Count how many heads you get. Count how many heads you get.

Repeat this trial 10 times.Repeat this trial 10 times.Create a histogram for your data Create a histogram for your data

(frequency of how many heads you (frequency of how many heads you got in each of the 10 trials).got in each of the 10 trials).

Put your histogram on the board.Put your histogram on the board.

Remember…Remember…

When we explore data on a single When we explore data on a single quantitative variable:quantitative variable:Plot your data (usually a histogram or Plot your data (usually a histogram or

stemplot)stemplot)Look for the overall pattern (center, Look for the overall pattern (center,

shape, and spread) and for outliersshape, and spread) and for outliersCalculate a numerical summary to Calculate a numerical summary to

describe center (median or mean) and describe center (median or mean) and spread (IQR or standard deviation)spread (IQR or standard deviation)

From Histograms to CurvesFrom Histograms to Curves

Sometimes, the overall pattern from a large Sometimes, the overall pattern from a large number of observations is so regular that number of observations is so regular that we can overlay a smooth curve.we can overlay a smooth curve.

Mathematical ModelsMathematical Models

This curve is a This curve is a mathematical modelmathematical model, , or an idealized description, for the or an idealized description, for the distribution. distribution.

It is easier to work with the smooth It is easier to work with the smooth curve than with the histogram.curve than with the histogram.

Density CurvesDensity Curves

Density curves are always positive (meaning Density curves are always positive (meaning it’s always on or above the horizontal axis).it’s always on or above the horizontal axis).

Areas under the curve represent proportions Areas under the curve represent proportions of the observations. of the observations. The area under a The area under a density curve always equals 1.density curve always equals 1.

The density curve describes the overall The density curve describes the overall pattern of a distribution. The area under the pattern of a distribution. The area under the curve, within a range of values, is the curve, within a range of values, is the proportion of all observations that fall in that proportion of all observations that fall in that range.range.

QuartilesQuartiles

How much area would be to the left How much area would be to the left of the first quartile?of the first quartile?

How much area would be to the right How much area would be to the right of the first quartile?of the first quartile?

How much area would be between How much area would be between the first and third quartiles?the first and third quartiles?

What Does All of This Mean?What Does All of This Mean?

When a density curve is a geometric When a density curve is a geometric shape (rectangle, trapezoid, or a shape (rectangle, trapezoid, or a combination of shapes) we can use combination of shapes) we can use geometry to find areas. Those areas geometry to find areas. Those areas help us find the median and the help us find the median and the quartiles.quartiles.

Verify that the graph is a valid density curve.Verify that the graph is a valid density curve. For each of the following, use areas under For each of the following, use areas under

density curve to find the proportion of density curve to find the proportion of observations within the given interval.observations within the given interval. 0.6<X<0.80.6<X<0.8 0<X<0.40<X<0.4 0<X<0.20<X<0.2

The median of this density curve is a point The median of this density curve is a point between X = 0.2 and X = 0.4. Explain between X = 0.2 and X = 0.4. Explain why.why.

I’m seeing Greek!I’m seeing Greek!

In a distribution, mean is x-bar and the In a distribution, mean is x-bar and the standard deviation is s.standard deviation is s.

When looking at a density curve, the mean When looking at a density curve, the mean is is μμ (pronounced mu) and the standard (pronounced mu) and the standard deviation is deviation is σσ (pronounced sigma). (pronounced sigma).

Normal DistributionsNormal Distributions

Density curves have an area = 1 and Density curves have an area = 1 and are always positive.are always positive.

Normal curves are a special type of Normal curves are a special type of density curves. Normal curves are density curves. Normal curves are symmetrical density curves.symmetrical density curves.T/F All density curves are normal T/F All density curves are normal

curves.curves.T/F All normal curves are density T/F All normal curves are density

curves.curves.

Characteristics of Normal Characteristics of Normal CurvesCurves

SymmetricSymmetricSingle-peaked Single-peaked

(also called (also called unimodal)unimodal)

Bell-shapedBell-shaped

The mean, μ, is located at the center of the curve.

The standard deviation, σ, is located at the inflection points of the curve.

μ

σ

Parameters of the Normal Parameters of the Normal CurveCurve

A normal curve is A normal curve is defined by its defined by its mean and standard mean and standard deviation.deviation.

Notation for a Notation for a normal curve is normal curve is N(N(μμ, , σσ).).

Why Be Normal?Why Be Normal?

Normal curves are good descriptions for Normal curves are good descriptions for lots of real data: SAT test scores, IQ, lots of real data: SAT test scores, IQ, heights, length of cockroaches (yum!).heights, length of cockroaches (yum!).

Normal curves approximate random Normal curves approximate random experiments, like tossing a coin many experiments, like tossing a coin many times.times.

Not all data is normal (or even Not all data is normal (or even approximately normal). Income data is approximately normal). Income data is skewed right.skewed right.

The Empirical RuleThe Empirical Rulea.k.a. 68-95-99.7 Rulea.k.a. 68-95-99.7 Rule

All normal distributions follow this All normal distributions follow this rule:rule:68% of the observations are within one 68% of the observations are within one

standard deviation of the meanstandard deviation of the mean95% of the observations are within two 95% of the observations are within two

standard deviations of the meanstandard deviations of the mean99.7% of the observations are within 99.7% of the observations are within

three standard deviations of the meanthree standard deviations of the mean

Yay, Math!Yay, Math!

IQ scores on the WISC-IV are normally IQ scores on the WISC-IV are normally distributed with a mean of 100 and a distributed with a mean of 100 and a standard deviation of 15.standard deviation of 15.

Going up one Going up one σσ and down one and down one σσ from 100 from 100 gives us the range from 85 to 115. 68% of gives us the range from 85 to 115. 68% of people have an IQ between 85 and 115.people have an IQ between 85 and 115.

95% of people have an IQ between ____ 95% of people have an IQ between ____ and ____.and ____.

99.7% of people have an IQ between ____ 99.7% of people have an IQ between ____ and ____.and ____.

Try ThisTry This

The heights of women aged 18 – 24 are The heights of women aged 18 – 24 are approximately normally distributed with a approximately normally distributed with a mean mean μμ = 64.5 inches and a standard = 64.5 inches and a standard deviation deviation σσ = 2.5 inches. = 2.5 inches.

Between what two heights do the middle Between what two heights do the middle 95% fall?95% fall?

The tallest 2.5% of women are taller than The tallest 2.5% of women are taller than what?what?

What is the percentile for a woman who is What is the percentile for a woman who is 64.5 inches tall?64.5 inches tall?

The army reports that the distribution of The army reports that the distribution of head circumference among male soldiers head circumference among male soldiers is approximately normal with mean 22.8 is approximately normal with mean 22.8 inches and standard deviation 1.1 inches.inches and standard deviation 1.1 inches.

What percent of soldiers have a head What percent of soldiers have a head circumference greater than 23.9 inches?circumference greater than 23.9 inches?

What percentile is this?What percentile is this? What percent of soldiers have a head What percent of soldiers have a head

circumference between 21.7 inches and circumference between 21.7 inches and 23.9 inches?23.9 inches?

HomeworkHomework

Chapter 2 # 15a, 25, 41-45Chapter 2 # 15a, 25, 41-45