1. 3.3 Measures of Position
Measures of location in comparison to the mean.
- standard scores
- percentiles
- deciles
- quartiles

2. Z score or Standard Score
90 on a music test vs 45 on an English test
The Z score tells us how many standard deviations a data value is above or below the mean.

3. Z score or Standard Score
The Z score tells us how many standard deviations a data value is above or below the mean.
-Subtract the mean from the value and divide by standard deviation
Z =
population
sample Z=

4. A student scores a 65 on a calc test that has a mean of 50 and a standard deviation of 10. She scored a 30 on a history test with a mean of 25 and standard deviation of 5. Compare her relative positions on the two tests.

5. Test 1: X = 38, X = 40 and s = 5
Test 2: X = 94, X = 100, s = 10

6. A set of data has a mean of 105 and a standard deviation of 8
A set of data has a mean of 105 and a standard deviation of 8. Find the data values that correspond to the following z scores.
a. 2
b. -1
c. 0
d. -1.6

7. Percentiles
Percentiles divide the data set into 100 equal groups

9. Percentiles are symbolized by P1,P2, P3, ...P100
Dividing the distribution into 100 groups.
P99 P1 P2 1% 1% 1%

10. Percentiles
The percentile corresponding to a given value X is found using the following formula:
(number of values below X) + .5
100%
Percentile =
total number of values

11. A teacher gives a 20 point test to 10 students
A teacher gives a 20 point test to 10 students. Find the percentile rank of the score of 12. Then find the percentile rank of the score of 6.
18,15,12,6,8,2,3,5,20,10

12. 18,15,12,6,8,2,3,5,20,10
Now use the data to determine the value corresponding with the 25th percentile.
n = # values
p = percentile
Formula to use: c = n*p
100
if c is not a whole number: round up to the nearest whole number
if c is a whole number find the value halfway between the cth term and the c+1 term

13. Find the value that corresponds to the 60th percentile
2,3,5,6,8,10,12,15,18,20

14. The frequency distribution for the systolic blood pressure readings (in mm or mercury) of 200 randomly selected college students is shown here. Construct a Percentile Graph.
Boundaries
Frequency
cumulative frequency
cumulative percent 24 62 72 26 12 4

15. Cumulative Percentages
Class Boundaries

16. Quartiles
Quartiles divide the distribution into 4 groups: Q1,Q2,Q3,
Q1 --> is the same as P25 or 25th percentile
Q2 --> is the same as P50 or 50th percentile
Q3 --> is the same as P75 or 75th percentile

17. MD
largest data value
smallest data value Q1 Q2 Q3
25%
25%
25%
25%

18. Finding the Quartiles
Q2 --> The Median! Q1
--> The median of the data below Q2 Q3
--> The median of the data above Q2

19. Find Q1,Q2 and Q3 for the following data set.
15,13,6,5,12,50,22,18

20. Deciles Deciles divide the data into ______ groups.
We can use the formula for Percentiles to find Deciles

21. Interquartile Range
Quartiles can be used as a rough measurement of variability
Interquartile Range: (IQR) the difference between Q3 and Q1. Or the range of the middle 50% of the data.
We can use the IQR to identify outliers

22. Outlier
An extremely high or extremely low value when compared to the rest of the data.
Outliers affect:
- mean
-standard deviation
-range

23. How do we determine if a value is high or low enough to be an outlier?
1. Put the data in order and find the quartiles.
2. Find IQR
3. Multiply the IQR by 1.5
4. Subtract the product from Q1 and add it to Q3.
5. If there are any values lower or higher than those two values, they are considered outliers.

24. Check the following for outliers
5,6,12,13,15,18,22

25. Using the Calculator 1. Enter data into L1 2. Press stat
3. Move the arrow 1 right to Calc
4. Press 1 for Var-Stats
5. Press 2nd L1 then enter

26. Using the Calculator Your calculator will display the following:
x: sample mean
x: sum of the data values
x2: sum of the squares of the data values
Sx: sample standard deviation
: population standard deviation
minX: smallest data value
Q1: lower quartile
Med: median
Q3: upper quartile
maxX: largest data value

27. Using the Calculator
Use your calculator to find the stats on the following data:
11.2, 11.9, 12.0, 12.8, 13.4, 14.3

28. Using the Calculator For grouped data... 1. Enter midpoints into L1
2. Enter frequencies into L2
3. Press stat button
4. Use arrow to move 1 right to calc
5. Press 1 for vars stats
6. Press 2nd L1 and 2nd L2 then enter

29. Using the Calculator
Find the mean and standard deviation of the following data. 1 8 2 13 3 18 5 23 4 28 33 38

30. Using the Calculator
Graph a percentile graph on your calculator 1 2 3 5 4