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Measures of Position Types of Measurements (Standard Score) • Standard Score: This score represents the number of standard deviations a data value falls above or below the mean. Symbol for standard score is z. • Positive z score is above the mean, negative is below the mean, and 0 is same score as the mean. • Obtained by taking a data value – mean then dividing by the standard deviation. Standard Score Example 1 • Test A: Score was 72. Mean was 68. Standard deviation was 5. • Test B: Score was 81. Mean was 80. Standard deviation was 2. • On which test is the relative position better? Types of Measurements (Percentiles) • Percentiles: Position measures normally used in the educational and health-related fields to indicate position of an individual in a group. • Percentiles are not the same as percentages. Example: If students gets an 88% on a test that does not indicate their position with respect to the rest of the class. If 88% corresponds to the 80th percentile than they did better than 80% of the students in the class. • Symbolized by P₁…P99 and divides the distribution up into 100 parts. Percentile Formula • The percentile corresponding to a given value (X) is computed by using the following formula: • Percentile = Number of values below +0.5 ● 100% Total number of values Percentile Examples • A teacher gives a 30 point test to 12 students. The scores are shown below. Find the percentile rank of a score of 21. (16,28,30,19,21,23,22,17,15,26,17,24) • Step 1: Arrange data in order from least to greatest. • Step 2: Next substitute in the formula. Finding values for a corresponding percentile. • Use the following set to find the value corresponding to the 40th percentile. (10,12,8,16,7,11,5,15,9,14) • Step 1: Lowest to greatest • Step 2: Compute using c = ( n ● p ) ÷ 100 • n = total number of values, p = percentile • Step 3: If c is not a whole number round to next whole number. If c is whole need to use average of c and c +1 value. Creating a percentile graph • Use the data to create a percentile graph. Class limit 19 - 25 26 - 32 33 - 39 40 - 46 47 - 53 Frequency 9 10 4 8 9 • Step 1: Create a cumulative freq. column and a cumulative percent column. • Step 2: Class boundaries create x-axis and percents create y-axis. Deciles, Quartiles, and Outliers • Deciles: divide the data into 10 groups. • Denoted by the symbol D1, D2,…D9 which correspond to P10, P20,…P90. • Quartiles: Divide the data into 4 groups. • Denoted by the symbol Q1, Q2, Q3 which correspond to P25, P50, P75. • Outliers: Is an extremely high or low value when compared to the rest of the data values.