Chapter 5. Normal. Normal Curve. the Normal. Curve Examples. Standard Units Standard Units Examples. for Data

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1 curve Approximation Part II Descriptive Statistics The Approximation Approximation The famous normal curve can often be used as an 'ideal' histogram, to which histograms for data can be compared. Its equation is y = 1 2π e x % but we will work with it through diagrams tables, without ever using the equation. curve Approximation It was discovered in 1720 by Abraham de Moivre, but is also called Gaussian curve or bell curve. Approximation The graph is symmetric about 0, the total area equals 100%. The curve is always above the horizontal axis. The area under the normal curve between -1 1 is about 68%.

2 curve curve Approximation The graph is symmetric about 0, the total area equals 100%. The curve is always above the horizontal axis. Approximation The graph is symmetric about 0, the total area equals 100%. The curve is always above the horizontal axis. The area under the normal curve between -2 2 is about 95%. The area under the normal curve between -3 3 is about 99.7%. Areas under the normal curve Other areas can be found using tables (page A104 in textbook). Approximation Approximation 1 Find the area between 0 1 under the normal curve. 2 Find the area to the right of 1 under the normal curve. 3 Find the area to the right of 0.45 under the normal curve.

3 units The HANES study Approximation Many histograms are similar in shape to the normal curve, provided they are drawn to the same. Making the horizontal s match up between histogram normal curve involves stard units. A value is converted to stard units by checking how many SDs it is above or below the average. Values above the average get a plus sign, values below the average get a minus sign. Approximation HANES: Health And Nutrition Examination Survey ( ) A representative cross-section of 20,322 Americans age 1 to 74 was examined. Data was obtained on demographic variables: age, education, income physiological variables: height, weight, blood pressure, cholesterol levels dietary habits levels of lead pesticides in the blood prevalence of diseases Example 1 Women age in the HANES study Approximation Average height is 63.5 inches SD is 2.5 inches inches = 63.5 }{{ inches } + 5} inches {{} average 2SD In stard units, this is inches = 63.5 }{{ inches } 2.5 } inches {{} average 1SD In stard units, this is -1. Approximation

4 68% - 95% rules 68% - 95% rules Approximation 68% rule For many lists 68% of the entries are between (average - SD) (average + SD). Where does this rule come from? convert the interval to stard units: -1 to 1 the area under the normal curve between -1 1 is 68% if the histogram follows the normal curve, the area under the histogram is about 68% Approximation 95% rule For many lists 95% of the entries are between (average - 2 SDs) (average + 2 SDs). Where does this rule come from? convert the interval to stard units: -2 to 2 the area under the normal curve between -2 2 is 95% if the histogram follows the normal curve, the area under the histogram is about 95% approximation for data Example 1, continued Approximation We have seen that many histograms are very similar to the normal curve if we draw them in stard units. Then the normal curve can be used to estimate the percentage of entries in an interval as follows: 1 convert the interval to stard units 2 nd the corresponding area under the normal curve Denition This procedure is called the normal approximation. Approximation Use the normal curve to estimate the percentage of women age in the HANES study sample who's height is between inches.

5 Example 2 (p. 88, Ex. Set C #2) Areas under the normal curve Approximation In a law school class, the entering students averaged about 160 on the LSAT; the SD was about 8. The histogram of LSAT scores followed the normal curve reasonably well. About what percentage of the class scored below 166? One student was 0.5 SDs above average on the LSAT. About what percentage of the students had lower scores? Approximation rank Approximation We can also summarize data by looking at percentiles. Denition The k-th percentile is a number such that k% of the entries in a list are smaller than the number, (100-k)% are larger. : 1st percentile = number such that 1% of the entries are smaller than the number, 99% are larger 25th percentile = number such that 25% of the entries are smaller than the number, 75% are larger Approximation Denition The percentile rank of a value is the percentage of entries smaller than that value. : the percentile rank of the highest homework score is 100% the percentile rank of the median homework score is 50%

6 Example 3: percentile vs. percentile Example 3: percentile vs. percentile Approximation rank percentile rank are each other's opposites! Table: Selected percentiles for family income in the U.S. in 2004 (see page 89 in textbook) 1 $0 10 $15, $29, $54, $90, $135, $430,000 Approximation Income data (table from previous slide): The 10th percentile is $15, 000 rank It is the number such that 10% of the entries are smaller than that number The percentile rank of $15, 000 is 10% It is the percentage of entries smaller than $15, 000 Another way of saying this: an annual income of $15, 000 puts you at the 10th percentile of the income distribution A percentile is a number A percentile rank is a percentage the normal curve Example 4 (p.92, Ex Set E #2) Approximation If a histogram follows the normal curve, then the normal curve can be used to estimate percentiles Method: 1 sketch a normal curve; nd the right value of z, using the normal table 2 z is given in stard units; convert it back to the units in the problem Approximation Among all applicants to a university, the Math SAT scores averaged 535, the SD was 100, the scores followed the normal curve. Estimate the 80th percentile.

7 Areas under the normal curve Example 5 (p. 92, Ex. Set E #3) Approximation Approximation For a Berkely freshmen, the average GPA is around 3.0; the SD is about 0.5. The histogram follows the normal curve. Estimate the 30th percentile of the GPA distribution. Approximation Denition 1st quartile = nr. such that 1/4 of the data are smaller 3/4 are larger = 25th percentile 2nd quartile = nr. such that 2/4 of the data are smaller 2/4 are larger = 50th percentile = median 3rd quartile = nr. such that 3/4 of the data are smaller 1/4 are larger = 75th percentile Approximation Denition Interquartile Range (IQR) is another measure of the spread of the data. It is given by Interquartile Range = 3rd quartile - 1st quartile

8 Motivation Eects of change of Approximation Changes of often occur: length: cm - inches - feet temperature: Fahrenheit - Celsius Such changes of consist of: multiplying all entries by a constant adding a constant to all entries both of the above How does this inuence the average, the SD the stard units? Approximation If we add a constant to all entries in a list, then the average increases by this constant the SD does not change the stard units do not change If we multiply all entries in a list by a positive number, then the average is multiplied by this number the SD is multiplied by this number the stard units do not change Approximation Example 6 (p. 93, Ex Set F #10) A group of people have average temperature of 98.6 F, with SD of 0.3 F. 1 Translate this to C. 2 How much is 1.5 SDs on Fahrenheit in Celcius? C = 5 (F 32 ) 9 Approximation How to summarize data that don't Don't use the normal curve! follow the normal curve? Good ways to summarize such data are: the histogram a table with percentiles (like for the income data) the 1st quartile, median 3rd quartile box-plot: box given by 1st 3rd quartile: contains the middle 50% of the data median is given as a line in the box it also gives some information on entries that fall outside the box

9 Is the normal approximation Approximation ' ( )' )( *' *(! " # $ % & Approximation reasonable? Long-left tail distribution? Heights of STAT 220 students? Daily Rainfall? Family Income? Length of time spent on STAT 220 Quiz 1? The Approximation is discussed again in Chapter 18. Long-left tail distribution? Heights of STAT 220 students? Approximation Density Points Approximation

10 Daily rainfall? Family income? Approximation...THE SEATTLE-TACOMA AIRPORT CLIMATE SUMMARY FOR THE MONTH OF SEPTEMBER WEATHER OBSERVED NORMAL DEPART LAST YEAR`S VALUE DATE(S) VALUE FROM VALUE NORMAL PRECIPITATION (INCHES) RECORD MAXIMUM MINIMUM T 1991 TOTALS DAILY AVG DAYS >=.01 5 MM MM 10 DAYS >=.10 2 MM MM 6 DAYS >=.50 1 MM MM 2 DAYS >= MM MM 1 GREATEST 24 HR. TOTAL /20 TO 09/ (From Approximation Table: Selected percentiles for family income in the U.S. in 2004 (see page 89 in textbook) 1 $0 10 $15, $29, $54, $90, $135, $430,000 Length of time spent on STAT 220 Is the normal approximation Quiz 1? reasonable? Approximation Approximation Long-left tail distribution? No! Heights of STAT 220 students? Not Bad. Daily rainfall? No! Family income? No! Length of time spent on STAT 220 Quiz 1? No!

11 Approximation Summary If a histogram has the shape of a normal curve, we can use the normal approximation to obtain information about the data behind the histogram based our knowledge of the normal curve. For this, the data must be converted to stard units., percentile ranks, quartiles are numerical summary information about our data that can be useful. Sometimes, we need to change the of the data. This might inuence the average the SD but not the stard units. approximation should only be used when the histogram follows the normal curve. The summary information can always be used.