Z-value SamplePopulation 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 and the standard deviation was A person with a BMI of 19.2 has a z score of: 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 30 th Percentile (P 30 ) if at least 30% of the data is below that value The 70 th Percentile (P 70 ) is a value for which 70% of the data is below that value What is P 50 ? The median (since 50% of the data is below the median)

Finding Percentiles To find what percentile a data value is in: Percentile of x = Number of values less than x Total number of values. 100 Example: In a class of 30 people, if you do better on a test than 24 other people, your percentile would be: You’re in the 80 th percentile

Finding a value from a Percentile Sort data Find locator k = percentile n = number of values If L is a whole number: The value of the k th percentile is between the L th 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 k th percentile is the L th 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 P 25 (25 th Percentile): Since L is not a whole number, round it up to 3. P 25 is the 3 rd data value, So P 25 = 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 P 75 (75 th Percentile): Since L is a whole number, we have to find the mean of the 6 th and 7 th data values (25.2 and 27.5). ( )/2=26.35 So P 75 = 26.35

5 number summary We want to summarize a data set with 5 numbers. min, __________, median, _________, max What should we use for these other two? P 25 = Q 1 P 75 = Q 3

Quartiles Q 1 = First Quartile = P 25 Q 2 = Second Quartile = P 50 = median Q 3 = Third Quartile = P 75 Note: Excel and your calculator can calculate Q 1 and Q 3, but there is not universal agreement on the procedure, and different tools with sometimes give different results.

Graphing the 5-number summary: The boxplot Q1Q1 Min Median Q3Q3 Max

How the Boxplot reveals the distribution

Using Boxplots to make Comparisons Males Females

Homework 2.6: 1, 3, 7, 13, 17, : 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.)