1 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!) Revised 2002 Statistics Show #3 of 3

2 The NORMAL Curve Properties & Example

3 Properties of the Normal Curve Bell Shaped (symmetric) distribution Mean = Median (  is the mathematical symbol used for this measure of center) The standard deviation(  ) is used for the measure of spread. THE 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. It is important to remember that this rule applies only to NORMALLY DISTRIBUTED DATA. Using the mean and standard deviation to describe the spread of the data can be done for any set of data. Just because you choose this measure of spread does not mean that the rule applies!

4 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....SET OF DATA… (a ranked list of all the values in the set) 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 set 50% of the data that is the same as or less than M. This value (M) is also called the 50th %ile M 95% of the data that is the same as or less than some data value, D. This value (D) is also called the 95th %ile D 5% of the data values

5 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......SET OF Normally distributed DATA… You know that M = 50%ile 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 Similarly, on the other side, 50% + 34% = 84%ile M Measure one s.d. to the left and one to the right of M. This is where you find 68% of the data. M 68% of the data 34% 50%ile 16%ile 84%ile Watch!

6 Using the properties of the Normal Curve... Bell Shaped (symmetric) distribution Mean = Median (  ) = 50th percentile The standard deviation(  ) is used for the measure of spread rule... DISTRIBUTION OF VALUES:  -3 .15%  -2  2.5%  -  16%  +  84%  + 2  97.5%  50%  + 3  99.85% You can use the rule to determine percentile ranks for other points on the Normal Curve... 68% 95% 99.7%

7 Normal Curve… common notation DISTRIBUTION OF VALUES: using STANDARD SCORES (Z-SCORES) (ex.) a standard score(z-score) of +1 = an actual score of  +  (ex.) a z-score of -3 = an actual score of  -3  Another common notation that is used when describing normally distributed data is called a STANDARD VALUE (standard score, z-score) This notation simply uses the standard deviation as a “standard unit, “ and describes how far away (in standard deviations) each actual data value is from the mean. The sign of the standard value indicates its direction from the mean % % 16% +1 84% % 0 50% %  -3  -2  -  +  + 2    + 3  Each actual data value corresponds to a standard value.

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 %2.5%16% 84%97.5% 50% 99.85% Put the actual values on the graph and complete the application Weights in pounds

9 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% 84%97.5% 50% 99.85% (a) A woman weighing 140 lb. will be in which percentile? (b) A woman weighing 122 lb. will be in which percentile? (c) A woman the 75th percentile will weigh approximately…? 75% 2.5th %ile 84th %ile About 137 pounds

10 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; therefore, the weight is between 134 and 140 lb Application Analysis There are charts that can help you estimate more accurately between the standard percentiles; but we won’t discuss them here.

11 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!

12 End of show #3 This is the final show in the Statistics Unit REVISED 2002 Prepared by Kimberly Conti, SUNY Fredonia Suggestions and comments to: