Measures of Position

● The standard deviation is a measure of dispersion that uses the same dimensions as the data (remember the empirical rule) ● The distance of a data value from the mean, calculated as the number of standard deviations, would be a useful measurement ● This distance is called the z-score

● If the mean was 20 and the standard deviation was 6  The value 26 would have a z-score of 1.0 (1.0 standard deviation higher than the mean)  The value 14 would have a z-score of –1.0 (1.0 standard deviation lower than the mean)  The value 17 would have a z-score of –0.5 (0.5 standard deviations lower than the mean)  The value 20 would have a z-score of 0.0

● The population z-score is calculated using the population mean and population standard deviation ● The sample z-score is calculated using the sample mean and sample standard deviation

● z-scores can be used to compare the relative positions of data values in different samples  Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12  Pat received a grade of 72 on her biology exam where the mean grade was 65 and the standard deviation was 10  Pat received a grade of 91 on her kayaking exam where the mean grade was 88 and the standard deviation was 6  Calculate each z-score and see what class has the highest RELATIVE grade.

● Statistics  Grade of 82  z-score of (82 – 74) / 12 =.67 ● Biology  Grade of 72  z-score of (72 – 65) / 10 =.70 ● Kayaking  Grade of 81  z-score of (91 – 88) / 6 =.50 ● Biology was the highest relative grade

 The median divides the lower 50% of the data from the upper 50%  The median is the 50 th percentile  If a number divides the lower 34% of the data from the upper 66%, that number is the 34 th percentile

 The quartiles are the 25 th, 50 th, and 75 th percentiles  Q 1 = 25 th percentile  Q 2 = 50 th percentile = median  Q 3 = 75 th percentile  Quartiles are the most commonly used percentiles  The 50 th percentile and the second quartile Q 2 are both other ways of defining the median

● Quartiles divide the data set into four equal parts ● The top quarter are the values between Q 3 and the maximum ● The bottom quarter are the values between the minimum and Q 1

 Quartiles divide the data set into four equal parts  The interquartile range (IQR) is the difference between the third and first quartiles IQR = Q 3 – Q 1  The IQR is a resistant measurement of dispersion

 Can we find the Quartiles with a Calculator?  Data  1,2,3,4,5,6,8,10,15,20

● Extreme observations in the data are referred to as outliers ● Outliers should be investigated ● Outliers could be  Chance occurrences  Measurement errors  Data entry errors  Sampling errors ● Outliers are not necessarily invalid data

● One way to check for outliers uses the quartiles ● Outliers can be detected as values that are significantly too high or too low, based on the known spread ● The fences used to identify outliers are  Lower fence = LF = Q 1 – 1.5  IQR  Upper fence = UF = Q  IQR ● Values less than the lower fence or more than the upper fence could be considered outliers

● Are there any outliers? 1, 3, 4, 7, 8, 15, 16, 19, 23, 24, 27, 31, 33, 54 ● Calculations (You can use your Calculator to find these!)  Q 1 = 7  Q 3 = 27  IQR = 20  Lower Fence = Q 1 – 1.5  IQR  Upper Fence = Q  IQR

 z-scores  Measures the distance from the mean in units of standard deviations  Can compare relative positions in different samples  Percentiles and quartiles  Divides the data so that a certain percent is lower and a certain percent is higher  Outliers  Extreme values of the variable  Can be identified using the upper and lower fences