Transcript Z-value
Z-value Sample xx z s Population z x The z-value tells us how many standard deviations above or below the mean our data value x is. Positive z-values are above the mean, Negative z-values are below the mean Z-value example For a sample of females, the mean BMI (body mass index) was 26.20 and the standard deviation was 6.57. A person with a BMI of 19.2 has a z score of: x x 19.2 26.20 1.07 z 6.57 s So this person has a BMI 1.07 standard deviations below the mean “Unusual” values Greater than +2 (2 above the mean) or Less than –2 (2 below the mean) Percentiles A data value is in the 30th Percentile (P30) if at least 30% of the data is below that value The 70th Percentile (P70) is a value for which 70% of the data is below that value What is P50? The median (since 50% of the data is below the median) Finding Percentiles To find what percentile a data value is in: Number of values less than x .100 Percentile of x = Total number of values Example: In a class of 30 people, if you do better on a test than 24 other people, your percentile would be: 24 100 80 30 You’re in the 80th percentile Finding a value from a Percentile Sort data Find locator k n L 100 k = percentile n = number of values If L is a whole number: The value of the kth percentile is between the Lth value and the next value. Find the mean of those values If L is not a whole number: Round L up. The value of the kth percentile is the Lth value. Example BMI values: (9 values) 19.6, 19.6, 21.4, 22.0, 23.8, 25.2, 27.5, 29.1, 33.5 To find P25 (25th Percentile): 25 9 L 2.25 100 Since L is not a whole number, round it up to 3. P25 is the 3rd data value, 21.4. So P25 = 21.4 Example BMI values: (8 values) 19.6, 19.6, 21.4, 22.0, 23.8, 25.2, 27.5, 29.1 To find P75 (75th Percentile): 75 8 L 6 100 Since L is a whole number, we have to find the mean of the 6th and 7th data values (25.2 and 27.5). (25.2+27.5)/2=26.35 So P75 = 26.35 5 number summary We want to summarize a data set with 5 numbers. P25 = Q1 median, _________, P75 = Q3 max min, __________, What should we use for these other two? Quartiles Q1 = First Quartile = P25 Q2 = Second Quartile = P50 = median Q3 = Third Quartile = P75 Note: Excel and your calculator can calculate Q1 and Q3, but there is not universal agreement on the procedure, and different tools with sometimes give different results. Graphing the 5-number summary: The boxplot BMI (Females) 0 10 Min 20 Q1 30 Median 40 Q3 50 Max How the Boxplot reveals the distribution Using Boxplots to make Comparisons BMI Males Females 0 10 20 30 40 50 Homework 2.6: 1, 3, 7, 13, 17, 37 2.7: 3, 9 Read Review Do Review Exercises: 1-8 (on question 1, feel free to only use part of the data for calculations, then look up the full answer in the back before doing the rest of the problems.)