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Revised 2002 Statistics Show #3 of 3 Message to the user... The most effective way to use a PowerPoint slide show is to go to “SLIDE SHOW” on the top of the toolbar, and choose “VIEW SHOW” from the pull down menu. OR, using the shortcut toolbar on the bottom left, choose the rightmost icon (“SLIDE SHOW”) Use the spacebar, enter key or mouse to move through the slide show. Use the backspace key to undo the last animation on a slide TEACHERS: If using this show as part of a lecture, it is helpful to go to “PRINT” in the “FILE” menu and use the drop down menu at the bottom left: “PRINT WHAT.” For some shows, printing the “OUTLINE VIEW” will be helpful; as well as printing particular slides to use as handouts. (Many shows will include sound… you may want to turn on your speakers!) 1 The NORMAL Curve Properties & Example 2 Properties of the Normal Curve It is important to (symmetric) remember thatdistribution this rule applies only to •Bell Shaped NORMALLY DISTRIBUTED DATA. •Mean = Median ( is the mathematical symbol used Using the mean and standard deviation to describe the spread for this measure of center) of the data can be done for any set of data. •The standard ) is used for thedoes measure of Just because you deviation( choose thismeasure of spread not spread. mean that the rule applies! THE 68-95-99.7 RULE •Approximately 68% of the data lies within 1 standard deviation of the mean •Approximately 95% of the data lies within 2 standard deviations of the mean •Approximately 99.7% of the data lies within 3 standard deviations of the mean. 3 A brief explanation...Percentile ranks… • A percentile rank indicates a place in any set of data where a certain percentage of the data falls AT or BELOW. • For example, the MEDIAN is the 50th percentile, • which means that 50% of the data values are EQUAL TO or LESS THAN that median. • It can also be said that the MEDIAN is greater than or equal to 50% of the data values in the set. M 95% the that dataisthat the same 50% of theofdata theissame as or as or less than some data value, D. ...SET OF ranked list95th of all the values in the This value (D) (M) is also called the %ile less than M. ThisDATA… value is(a also called the 50th %ile D set) 5% of the data values • Another example… • The 95th percentile (95%ile) for a set of data would be... • that data value that is greater than or equal to 95% of all of the data values in the set. • Notice, also that it is less than or equal to 5% of the values in the set4 Percentile ranks and the Normal Distribution… • So, if you know that one property of a Normal Distribution is that 68% of the data lies within 1 standard deviation from the mean... • And you know that the mean = median for this type of distribution... • You can say something about percentile ranks for this set of data... 50%ile ...SET OF NormallyM distributed DATA… M Measure one s.d. to the left 34% 34% data and 68% one toof thethe right of M. This is where you find 68%84%ile • You know that M =16%ile 50%ile of the data. • and you know that a NORMAL DISTRIBUTION means that the data is SYMMETRIC about that data value… • So, 34% of the data is in that part of the set of values that is one standard deviation to the left of M • and 34% is in the set to the right of M • To find the percentile rank for the value to the LEFT of the mean, subtract: 50% -34% = 16%ile Watch! • Similarly, on the other side, 50% + 34% = 84%ile 5 Using the properties of the Normal Curve... • Bell Shaped distribution rule to determine You can use(symmetric) the 68-95-99.7 • Mean = Median () = 50th percentile percentile ranks for other points on the Normal • The standard deviation() is used for the measure of spread. • 68-95-99.7 rule... Curve... DISTRIBUTION OF VALUES: .15% 2.5% 16% 50% 84% 97.5% 99.85% 68% 95% 99.7% -3 -2 - + + 2 + 3 6 Normal Curve… common notation Another common notation that is used when describing normally DISTRIBUTION OF VALUES: distributed data is called a STANDARD VALUE (standard score, z-score) using STANDARD SCORES (Z-SCORES) This notation simply uses the standard deviation as a “standard unit, “ • describes (ex.) a standard score(z-score) of and how far away (in standard deviations) each actual data +1 = an actual score of + value is from the mean. The sign ofathe standard its direction from the mean. • (ex.) z-score ofvalue -3 =indicates an actual score of -3 .15% 2.5% 16% 50% 84% 97.5% 99.85% -3 -2 -1 -3 -2 - 0 +1 +2 +3 + + 2 + 3 Each actual data value corresponds to a standard value. 7 Application of the Normal Curve DISTRIBUTION OF VALUES for weights of adult women whose height is approximately 5ft. 5 in.: = 134 pounds .15% 2.5% 16% -3 116 = 6 pounds 50% -2 -1 0 122 128 134 84% 97.5% 99.85% +1 140 +2 146 +3 152 Weights in pounds Put the actual values on the graph and complete the application 8 Application of the Normal Curve DISTRIBUTION OF VALUES for weights of adult women whose height is approximately 5ft. 5 in.: = 134 pounds = 6 pounds .15% 2.5% 16% 50% 84% 97.5% 99.85% 75% 116 122 128 134 140 146 152 (a) A woman weighing 140 lb. will be in which percentile? 84th %ile (b) A woman weighing 122 lb. will be in which percentile? 2.5th %ile (c) A woman the 75th percentile will weigh About 137 pounds approximately…? 9 Application Analysis • DISTRIBUTION OF VALUES for weights of adult women whose height is approximately 5ft. 5 in.: = 134 pounds =6 pounds • (a) A woman weighing 140 lb. will be in the 84th percentile: 84 percent of women in this height category weigh 140 lb or less. (also, 16% weigh more than 140) • (b) A woman weighing 122 lb. will be in the 2.5th percentile: 2.5 percent of women weigh 122 lb or less.(and 97.5% weigh more) • (c) A woman in the 75th percentile will weigh approximately…? The 75th percentile is between the 50th and 84th; There are charts that can help you estimate more therefore, the weight is between 134 and 140 lb accurately between the standard percentiles; but we won’t discuss them here. 10 Analysis Using the Normal Distribution... • Many distributions that occur naturally exhibit a normal (or nearly normal) tendency… – – – – Weights of children Weights of adults in certain height categories Scores on state-wide or national tests etc… • The properties exhibited by normally distributed data allow us to make predictions about similar sets of data, • and to draw conclusions about these sets without having to handle each piece of data. • See your text for more examples of normal distributions and the uses of their properties! 11 End of show #3 This is the final show in the Statistics Unit REVISED 2002 Prepared by Kimberly Conti, SUNY College @ Fredonia Suggestions and comments to: [email protected] 12