Z SCORES PERCENTILES AND QUARTILES z Score Percentiles

Z – SCORES, PERCENTILES, AND QUARTILES

� � � z – Score Percentiles Quartiles

z - Score � � � A standardized value A number of standard deviations a given value, x, is above or below the mean z = (score (x) – mean)/s (standard deviation) A positive z-score means the value lies above the mean A negative z-score means the value lies below the mean Round to 2 decimals

z – Score Example � � � Score (x) = 130 Mean = 100 s = 15 z = (score (x) – mean)/s (standard deviation) z = (130 - 100)/15 (standard deviation) = 30/15, = 2 o The score of 130 lies 2 standard deviations above the mean (positive z means above the mean) o

z – Score Example � � � Score (x) = 85 Mean = 100 s = 15 z = (score (x) – mean)/s (standard deviation) z = (85 - 100)/15 (standard deviation) = -15/15, = -1 o The score of 85 lies 1 standard deviation below the mean (negative z means below the mean) o

Percentiles � � Measures of location which divide a set of data into 100 groups with about 1% of values in each group P 1, P 2, P 3, P 4, …P 99 Percentile of value x = number of values < x divided by the total number of values * 100 (for percent) Round to nearest whole number

Percentiles Example � � Percentile of value x = number of values < x divided by the total number of values * 100 (for percent) 5 6 6 7 9 10 15 17 16 16 18 20 22 24 25 27 30 31 Find the percentile for the value of 18

Percentiles Example � � Find the percentile for the value of 18 Percentile of 18 = 10 (numbers less than 18) 18 (total number of values) Percentile of 18 =. 55556 * 100 = 56% This means that the value of 18 is the 56 th percentile

Percentiles Example � � Converting a percentile into a data value L = the locator that gives the position of the value k = percentile Find the 20 th percentile, P 20 5 6 6 7 9 10 15 17 16 16 18 20 22 24 25 27 30 31

Percentiles Example � � � � Find the 20 th percentile, P 20 Compute L L = k/100 * n L = 20/100 * 18 L =. 20 * 18 = 3. 6 When L is not a whole number, round up instead of off L = the 4 th value, which is 7 in the table

Percentiles Example � � L = the 4 th value, which is 7 in the table This means that the 20 th percentile is the value 7 and about 20% of the values are below the value 7 and about 80% of the values are above the value 7. 5 6 6 7 9 10 15 17 16 16 18 20 22 24 25 27 30 31

Percentiles Example � � Converting a percentile into a data value L = the locator that gives the position of the value k = percentile Find the 50 th percentile, P 50 5 6 6 7 9 10 15 17 16 16 18 20 22 24 25 27 30 31

Percentiles Example � � � Find the 50 th percentile, P 50 Compute L L = k/100 * n L = 50/100 * 18 L =. 50 * 18 = 9 When L is a whole number, the value of the k percentile is midway between the Lth value and the next sorted value (take the average of the two values).

Percentiles Example � � � L = the 9 th value, which is 16 in the table Take this value plus the next sorted value, which is also 16, and calculate the average Here, the 9 th percentile is 16. 5 6 6 7 9 10 15 17 16 16 18 20 22 24 25 27 30 31

Quartiles � � � Measures of location which divide a set of data into four groups with about 25% of the values in each group Q 1, Q 2, Q 3, Q 4 Q 1 = P 25 = First quartile, the bottom 25% Q 2 = P 50 = Second quartile, same as the median Q 3 = P 75 = Third quartile, the upper 25%

Quartiles Example � � � Finding Q 1 is the same as finding P 25 L = k/100 * n L = 25/100 * 18 L =. 25 * 18 = 4. 5, or the 5 th value In the table, the 5 th value is 9 So, Q 1 = 9 5 6 6 7 9 10 15 17 16 16 18 20 22 24 25 27 30 31