Unit 3 Section 3-4

3-4: Measures of Position Measures of position are used to locate the position of a data value within a data set. Measures of Position Standard Score (z score) Percentiles Deciles Quartiles

Section 3-4 Standard Score – a value obtained by subtracting the mean from the value and dividing the result by the standard deviation. Also known as a z score Symbol: z Represents the number of standard deviations that a data value falls above or below the mean. Positive values are above the mean, negative are below the mean.

Standard Scores Section 3-4 A student scored a 65 on a calculus test that had a mean of 50 and a standard deviation of 10; she scored 30 on a history test with a mean of 25 and a standard deviation of 5. Compare her relative position on the two tests. Find the standard score for calculus Find the standard score for history The larger the z score, the higher her position in that class.

Standard Scores Section 3-4 Find the z score for each test. State which is higher. Test Value Mean Standard Deviation Test A 38 40 5 Test B 94 100 10

Section 3-4 Percentiles– divide the data set into 100 equal groups. Ranks a data value based on its position within the data set. Example: a percentile of 80% means that the specific data value is higher than 80% of the values within the data set. Percentiles often used to rank performance on individual tests such as PSAT

Finding the Percentile Section 3-4 Finding the Percentile Arrange the data in order from lowest to highest. Locate the corresponding data value. Count the number of values smaller than the corresponding value. Take that value, add 0.5 Divide by the total number of values. Multiply by 100 The result is the data value’s position as a percent.

Finding a Percentile Section 3-4 A teacher gives a 20-point test to 10 students. The scores are shown below. 18, 15, 12, 6, 8, 2, 3, 5, 20, 10 a) Find the percentile rank of a score of 12. b) Find the value corresponding to the 25th percentile.

Section 3-4 Quartiles– divide the data set into 4 equal groups. The quartiles are separated by the values Q1, Q2, and Q3. Q1 - 25th percentile Q2 – 50th percentile, or the median Q3 – 75th percentile Interquartile Range (IQR) – the difference between Q3 and Q1.

Finding the Quartile Section 3-4 Arrange the data in order from lowest to highest. Find the median of the data. (Q2) Find the median of the first half of the data. (Q1) Find the median of the second half of the data. (Q3)

Finding a Quartile Section 3-4 Find Q1, Q2, and Q3 for the data set: 15, 13, 6, 5, 12, 50, 22, 18

Section 3-4 Deciles– divide the data set into 10 equal groups. The deciles are labels as D1, D2, D3, … , D9. D1 - 10th percentile D2 – 20th percentile D3 – 30th percentile … Outlier – an extremely high or an extremely low data value when compared with the rest of the data values.

Finding an Outlier Section 3-4 Arrange the data in order to find Q1 and Q3. Find the IQR (Q3 – Q1). Multiply the IQR by 1.5 Lower Boundary: subtract that value from Q1 Upper Boundary: add that value to Q3. Check the data set for any value that is outside the boundaries.

Finding the Outlier Section 3-4 Check the following data set for outliers. 5, 6, 12, 13, 15, 18, 22, 50

Section 3-4 Homework Pg 144: 1-9, 13