HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? What is the 50 th percentile for the cigarette histogram?

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1 HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? The point on the horizontal axis such that of the area under the histogram lies to the left of that point (and to the right) What is the 25 th percentile in this case? % per cigarette (5) (35) (5) Distribution of the number of cigarettes smoked per day by male current smokers in Number of cigarettes (5) Point on horizontal axis Area to the left of it 5% 5% + 35% 2 5% + 7% 3 5% + 5% So the 25 th percentile is about 3 What does that say about the people in the study? out of 4 of them smoked 3 or fewer cigarettes per day Other percentiles are defined in a similar way Eg,, the 95 th percentile is the point on the horizontal axis such that 95% of the area under the histogram lies to the left of it % per cigarette What is the 5 th percentile for the cigarette histogram? (5) (35) (5) Number of cigarettes 5 th percentile = 2 (5) out of of these men smoked or fewer cigarettes per day The 25 th, 5 th, and 75 th percentiles are called quartiles: 25 th percentile = first quartile (Q) 5 th percentile = second quartile (2Q) = the median 75 th percentile = third quartile (3Q) The interquartile range (IQR) is the distance between the first and third quartiles; this is measure of spread that is not sensitive to outlying values In the cigarette example, Q 3 3Q 37 IQR = 3Q Q 37 3 =

2 WHY ARE NORMAL CURVES OF INTEREST? Normal curves often provide a simple, compact way of describing how some variable is distributed Many variables (eg, height, blood pressure,, but not years of education, ) have histograms which follow (match up well with) a normal curve: Histogram of heights of women in HANES (976 98) Approximating normal curve For such variables, areas under the histogram that is, population percentages can be approximated by the corresponding areas under the normal curve: HEIGHT (INCHES) Areas under the normal curve can be computed easily knowing only the average and the SD 5 3 Normal curves are well known and well understood A convenient means of communication As Chapter 8 explains, the sampling distribution of sample averages tends to follow the normal curve This is the cornerstone of statistical inference! THE STANDARD NORMAL CURVE STANDARD UNITS 2 4 PERCENT PER STANDARD UNIT The equation of the curve is ordinate = % 2π e (abscissa)2 /2 Two very important properties are: The total area under the curve is % Just like a histogram The curve is symmetric about 5 4

3 A BRIEF TABLE OF AREAS UNDER THE STANDARD NORMAL CURVE The following figure shows some benchmark areas under the standard normal curve: % 68% 95 2 % 95% % All but /4 of % What is the area under the standard normal curve to the right of? = half of = 2 = 2 [ % 68% ] = 2 32% = 6% 5 5 What is the area under the standard normal curve between and 2? 2 = = 2 [ 95% 68% ] = 2 27% = 3 2 % Alternatively 2 = 2 = 6% 2 2 % = 3 2 % What is the area under the standard normal curve between and 2? 2 = + 2 = 2 68% % = 34% % = 8 2 % 5 6

4 A NORMAL TABLE The following table is like the one on page A86 of FPPA, except that it omits the columns of heights of the normal curve: z z Area (percent) z Area z Area z Area Benchmarks z Area 5% 9% 95% THE QUARTILES OF THE STANDARD NORMAL CURVE What is the first quartile of the standard normal curve?? Area 75%?? =? Area % Area % Area % The quartiles of the standard normal curve are: Q = 675 2Q = 3Q = 675 The interquartile range (IQR) for the standard normal curve is IQR = 3Q Q = 675 ( 675) = /3 5 8

5 What are standard units? STANDARD UNITS Standard units say how many SDs a value is above (+ sign) or below ( sign) average The women in the HANES study had heights averaging to 635 inches, with an SD of 25 inches What is 6 in standard units? 6 is inches average That s SD average So 6 is in standard units What is 685 in standard units? 685 is inches average That s SDs average So 685 is in standard units What height is 24 in standard units? The height is SDs average That s = 6 inches average The height is Reminder: standard units say how many SDs a value is above (+ sign) or below ( sign) average Height (Inches) Standard Units (Dimensionless) Average = 635 SD = 25 Average = SD = Is there a formula for converting a value to standard units? Yes, the formula is standard units = value average SD In our example, to express 685 in standard units you compute = 5 25 = 2 Is there a formula for converting back from standard units to the original scale? Yes, the formula is Height (Inches) Standard Units (Dimensionless) Average = 635 SD = 25 value = average + (standard units SD) In our example, to find the height corresponding to 24 standard units, you compute ( ) = =

6 THE NORMAL APPROXIMATION If a list of numbers follows the normal curve, the percentage of entries falling in a given interval can be estimated by first converting the interval to standard units and then finding the corresponding area under the standard normal curve This procedure is called the normal approximation Consider the heights of women in HANES: USING THE NORMAL APPROXIMATION A group of people have heights that follow a normal curve with average 69 and SD 3 About what percentage of these people have heights 66 or under?? Height (inches) Ave = 69, SD = 3 Height (Inches) Standard Units (Dimensionless) Average = 635 SD = 25 Standard units The percentage of women with heights between 6 and 685 is exactly equal to the area under the from to, and approximately equal to the area under the between and, namely 85% If a histogram follows the normal curve, about percent of the area lies within one SD of the average, and about percent within two SDs of the average Warning: The normal approximation, especially for onesided areas, is only valid if the histogram is approximately normal Use your judgement 5 Answer = 6% (see page 5) The method: original units standard units standard normal curve Are these men, or women? Ave height = 5 foot 9: they re men 5 2

7 Same population (heights averaging to 69 with an SD of 3 ) What height is exceeded by 5% of the population? SUMMARY What is the general procedure for working these kinds of problems? DRAW THE PICTURE 69? 5% Height (inches) Ave = 69, SD = 3 Standard units Sketch the normal curve Put in the axis for the original units Put in the axis for the standard units Shade the area of interest Proceed Thus? = height = SDs above average = = = 74 = 7 2 Ave Name? (units?) Ave =?, SD =? The method: standard normal curve standard units original units Standard units Be sure to follow this procedure on the homework, quizzes, and exams!

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