Section 2.1 Density Curves

Get out a coin and flip it 5 times. Count how many heads you get. Get out a coin and flip it 5 times. Count how many heads you get. Repeat this trial 10 times. Repeat this trial 10 times. Create a histogram for your data (frequency of how many heads you got in each of the 10 trials). Create a histogram for your data (frequency of how many heads you got in each of the 10 trials). Put your histogram on the board. Put your histogram on the board.

Remember… When we explore data on a single quantitative variable: Plot your data (usually a histogram or stemplot) Plot your data (usually a histogram or stemplot) Look for the overall pattern (center, shape, and spread) and for outliers Look for the overall pattern (center, shape, and spread) and for outliers Calculate a numerical summary to describe center (median or mean) and spread (IQR or standard deviation) Calculate a numerical summary to describe center (median or mean) and spread (IQR or standard deviation)

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

Mathematical Models This curve is a mathematical model, or an idealized description, for the distribution. This curve is a mathematical model, or an idealized description, for the distribution. It is easier to work with the smooth curve than with the histogram. It is easier to work with the smooth curve than with the histogram.

Density Curves Density curves are always positive (meaning it’s always on or above the horizontal axis). Density curves are always positive (meaning it’s always on or above the horizontal axis). Areas under the curve represent proportions of the observations. The area under a density curve always equals 1. Areas under the curve represent proportions of the observations. The area under a density curve always equals 1. The density curve describes the overall pattern of a distribution. The area under the curve, within a range of values, is the proportion of all observations that fall in that range. The density curve describes the overall pattern of a distribution. The area under the curve, within a range of values, is the proportion of all observations that fall in that range.

Quartiles How much area would be to the left of the first quartile? How much area would be to the left of the first quartile? How much area would be to the right of the first quartile? How much area would be to the right of the first quartile? How much area would be between the first and third quartiles? How much area would be between the first and third quartiles?

What Does All of This Mean? When a density curve is a geometric shape (rectangle, trapezoid, or a combination of shapes) we can use geometry to find areas. Those areas help us find the median and the quartiles. When a density curve is a geometric shape (rectangle, trapezoid, or a combination of shapes) we can use geometry to find areas. Those areas help us find the median and the 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 density curve to find the proportion of observations within the given interval. For each of the following, use areas under density curve to find the proportion of observations within the given interval. 0.6<X<0.8 0<X<0.4 0<X<0.2

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

I’m seeing Greek! In a distribution, mean is x-bar and the standard deviation is s. In a distribution, mean is x-bar and the standard deviation is s. When looking at a density curve, the mean is μ (pronounced mu) and the standard deviation is σ (pronounced sigma). When looking at a density curve, the mean is μ (pronounced mu) and the standard deviation is σ (pronounced sigma).

Normal Distributions Density curves have an area = 1 and are always positive. Density curves have an area = 1 and are always positive. Normal curves are a special type of density curves. Normal curves are symmetrical density curves. Normal curves are a special type of density curves. Normal curves are symmetrical density curves. T/F All density curves are normal curves. T/F All density curves are normal curves. T/F All normal curves are density curves. T/F All normal curves are density curves.

Characteristics of Normal Curves Symmetric Symmetric Single-peaked (also called unimodal) Single-peaked (also called unimodal) Bell-shaped Bell-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 Curve A normal curve is defined by its mean and standard deviation. A normal curve is defined by its mean and standard deviation. Notation for a normal curve is N(μ, σ). Notation for a normal curve is N(μ, σ).

Why Be Normal? Normal curves are good descriptions for lots of real data: SAT test scores, IQ, heights, length of cockroaches (yum!). Normal curves are good descriptions for lots of real data: SAT test scores, IQ, heights, length of cockroaches (yum!). Normal curves approximate random experiments, like tossing a coin many times. Normal curves approximate random experiments, like tossing a coin many times. Not all data is normal (or even approximately normal). Income data is skewed right. Not all data is normal (or even approximately normal). Income data is skewed right.

The Empirical Rule a.k.a Rule All normal distributions follow this rule: All normal distributions follow this rule: 68% of the observations are within one standard deviation of the mean 68% of the observations are within one standard deviation of the mean 95% of the observations are within two standard deviations of the mean 95% of the observations are within two standard deviations of the mean 99.7% of the observations are within three standard deviations of the mean 99.7% of the observations are within three standard deviations of the mean

Yay, Math! IQ scores on the WISC-IV are normally distributed with a mean of 100 and a standard deviation of 15. IQ scores on the WISC-IV are normally distributed with a mean of 100 and a standard deviation of 15. Going up one σ and down one σ from 100 gives us the range from 85 to % of people have an IQ between 85 and 115. Going up one σ and down one σ from 100 gives us the range from 85 to % of people have an IQ between 85 and % of people have an IQ between ____ and ____. 95% of people have an IQ between ____ and ____. 99.7% of people have an IQ between ____ and ____. 99.7% of people have an IQ between ____ and ____.

Try This The heights of women aged 18 – 24 are approximately normally distributed with a mean μ = 64.5 inches and a standard deviation σ = 2.5 inches. The heights of women aged 18 – 24 are approximately normally distributed with a mean μ = 64.5 inches and a standard deviation σ = 2.5 inches. Between what two heights do the middle 95% fall? Between what two heights do the middle 95% fall? The tallest 2.5% of women are taller than what? The tallest 2.5% of women are taller than what? What is the percentile for a woman who is 64.5 inches tall? What is the percentile for a woman who is 64.5 inches tall?

The army reports that the distribution of head circumference among male soldiers is approximately normal with mean 22.8 inches and standard deviation 1.1 inches. The army reports that the distribution of head circumference among male soldiers is approximately normal with mean 22.8 inches and standard deviation 1.1 inches. What percent of soldiers have a head circumference greater than 23.9 inches? What percent of soldiers have a head circumference greater than 23.9 inches? What percentile is this? What percentile is this? What percent of soldiers have a head circumference between 21.7 inches and 23.9 inches? What percent of soldiers have a head circumference between 21.7 inches and 23.9 inches?

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