Kenwood Academy
AP Statistics
Chapter 2 - The Normal Distribution
| Describing Location in a |Objective: |
|Distribution |MEASURE position using percentiles |
| |INTERPRET cumulative relative frequency graphs |
| |MEASURE position using z-scores |
| |TRANSFORM data |
|Measuring Position: Percentiles |DEFINE and DESCRIBE density curves |
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|Examples: |The pth percentile of a distribution is the value with p percent of the observations less than it. |
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| |Here are the scores of all 25 students in Mr. Pryor’s statistics class on their first test: |
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| |79 81 80 77 73 83 74 93 78 80 75 67 73 |
| |77 83 86 90 79 85 83 89 84 82 77 72 |
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| |Problem: Use the scores on Mr. Pryor’s test to find the percentiles for the for the following students (how did they perform |
| |relative to their classmates): |
| |a) Jenny, who earned an 86. b) Norman, who earned a 72. |
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| |c) Katie, who earned a 93. d) the two students who earned scores of 80. |
|Cumulative Relative Frequency | |
|Graphs | |
| |A cumulative relative frequency graph displays the cumulative relative frequency of each class of a frequency distribution. |
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| |Age of First 44 Presidents When They Were Inaugurated |
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| |Age |
| |Frequency |
| |Relative frequency |
| |Cumulative frequency |
| |Cumulative relative frequency |
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| |40-44 |
| |2 |
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| |45-49 |
| |7 |
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| |50-54 |
| |13 |
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| |55-59 |
| |12 |
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| |60-64 |
| |7 |
|Example: | |
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| |65-69 |
| |3 |
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|Measuring Position: z-Scores | |
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| |Was Barack Obama, who was inaugurated |
| |at age 47, unusually young? |
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| |Was Ronald Reagan, who was inaugurated |
| |at age 69, unusually old? |
|Example: | |
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|Transforming Data | |
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|Effect of Adding (or Subtracting) a| |
|Constant | |
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|Effect of Multiplying (or Dividing)|Check Your Understanding page 89 |
|by a Constant | |
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|Example: | |
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|Remember: Exploring Quantitative | |
|Data | |
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| |A z-score tells us how many standard deviations from the mean an observation falls, and in what direction. |
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| |To compare data from distributions with different means and standard deviations, we need to find a common scale. We accomplish |
|Density Curves |this by using standard deviation units (z-scores) as our scale. Changing to these units is called standardizing. Standardizing |
| |data shifts the data by subtracting the mean and rescales the values by dividing by their standard deviation. |
| |[pic] or [pic][pic] |
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| |Standardizing does not change the shape of the distribution. It changes the center (shifts it to zero) and the spread by making |
| |the standard deviation one. |
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| |Check Your Understanding page 91 |
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|Median and Mean of a Density Curve | |
| |Transforming converts the original observations from the original units of measurements to another scale. Transformations can |
| |affect the shape, center, and spread of a distribution. |
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| |Adding the same number a (either positive, zero, or negative) to each observation: |
| |adds a to measures of center and location (mean, median, quartiles, percentiles), but |
| |Does not change the shape of the distribution or measures of spread (range, IQR, standard deviation). |
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|Example: | |
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| |Multiplying (or dividing) each observation by the same number b (positive, negative, or zero): |
| |multiplies (divides) measures of center and location by b |
| |multiplies (divides) measures of spread by |b|, but |
| |does not change the shape of the distribution |
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| |Check Your understanding page 97 |
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|2.2 Normal Distribution |To describe a distribution: |
| |Make a graph |
| |Look for overall patterns (shape, center, and spread) and outliers |
| |Calculate a numerical summary to describe the center (mean, median) and spread (minimum, maximum, Q1, Q3, range, IQR, standard |
| |deviation) |
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|Normal Distributions |In addition to the above distributions sometimes the overall pattern of a large number of observations is so regular that we can |
|N([pic],[pic]) |describe it by a smooth curve. |
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| |A density curve describes the |
| |overall pattern of a distribution |
| |Is always on or above the |
| |horizontal axis |
| |Has exactly 1 underneath it |
| |The area under the curve and |
| |above any range of values is the |
| |proportion of all observations |
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|The 68-95-99.7 Rule | |
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| |The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of|
| |the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars. |
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| |Median of a density curve is the equal areas point, the point that divides the are under the curve in half |
| |Mean of a density curve is the balance point, at which the curve would balance if made of solid material. |
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|Application of the | |
|68-95-99.7 Rule | |
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| |Check Your Understanding page 103 |
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|The Standard Normal Distribution |Objectives: |
| |DESCRIBE and APPLY the 68-95-99.7 Rule |
| |DESCRIBE the standard Normal Distribution |
| |PERFORM Normal distribution calculations |
| |ASSESS Normality |
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|Z-Score Table | |
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| |All Normal curves have the same overall shape: symmetric, single-peaked, bell shaped. |
| |A Normal distribution is described by a Normal density curve. |
|Example |A Normal distribution can be fully described by two parameters, its mean μ and standard deviation σ |
| |The mean, µ, of a Normal distribution is at the center of the symmetric Normal curve and is the same as the median. |
| |The standard deviation σ controls the spread of a Normal curve. Curves with larger standard deviations are more spread out. |
|4-Step Process |The standard deviation, σ, is the distance from the center to the change-of-curvature points on either side. |
| |A short-cut notation for the normal distribution in N(μ,(). |
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| |All normal curves obey the 68-95-99.7% (Empirical) Rule. |
| |This rule tells us that in a normal distribution approximately |
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|Normal calculations | |
|Example: |68% of the data values fall within one standard |
| |deviation (1σ) of the mean, |
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| |95% of the values fall within 2σ of the mean, and |
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| |99.7% (almost all) of the values fall |
| |within 3σ of the mean. |
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|Using Table A in Reverse | |
| |Distribution of the heights of young women aged 18 to 24 |
| |What is the mean μ? |
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| |What is the (? |
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| |What is the height range for 95% of young women? |
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| |What is the percentile for 64.5 in.? |
|z-scores on the calculator | |
| |What is the percentile for 59.5 in.? |
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| |What is the percentile for 67 in.? |
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|Assessing Normality |What is the percentile for 72 in.? |
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| |The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. |
| |If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable |
| |[pic][pic] has the standard Normal distribution, N(0,1). |
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|Normal Probability Plots | |
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| |Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table. |
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| |Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the|
| |left of z. |
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| |Check Your Understanding page 119 |
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| |How to Solve Problems Involving Normal Distributions |
| |State: Express the problem in terms of the observed variable x. |
| |Plan: Draw a picture of the distribution and shade the area of interest under the curve. |
| |Do: Perform calculations. |
| |Standardize x to restate the problem in terms of a standard Normal variable z. |
| |Use Table A and the fact that the total area under the curve is 1 to find the required area under the standard Normal curve. |
| |Conclude: Write your conclusion in the context of the problem. |
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| |Women’s heights are approximately normal with N(64.5, 2.5). What proportion of all young women are less than 68 inches tall? |
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| |On the driving range, Tiger Woods practices his swing with a particular club by hitting many, many balls. When tiger hits his |
| |driver, the distance the balls travel follows a Normal distribution with mean 304 yards and standard deviation 8 yards. What |
| |percent of Tiger’s drives travel at least 290 yards? |
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| |What percent of Tiger’s drives travel between 305 and 325? |
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| |High levels of cholesterol in the blood increase the risk of heart disease. For 14 year old boys, the distribution of blood |
| |cholesterol is approximately Normal with mean µ = 170 milligrams of cholesterol per deciliter of blood (mg/dl) and standard |
| |deviation σ = 30 mg/dl. What is the first quartile off the distribution of blood cholesterol? |
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| |Technology Corner page 123 |
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| |Plot the data. |
| |Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell-shaped. |
| |Check whether the data follow the 68-95-99.7 rule. |
| |Count how many observations fall within one, two, and three standard deviations of the mean and check to see if these percents |
| |are close to the 68%, 95%, and 99.7% targets for a Normal distribution. |
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| |If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal. Systematic |
| |deviations from a straight line indicate a non-Normal distribution. Outliers appear as points that are far away from the overall |
| |pattern of the plot. |
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| |Example page 127 |
| |Technology Corner page 128 |
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|Summary |
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