3.3 Measures of Position

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

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.

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 =

Z=

population

sample

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.

Test 1: X = 38, X = 40 and s = 5

Test 2: X = 94, X = 100, s = 10

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. 2b. -1c. 0d. -1.6

Percentiles

Percentiles divide the data set into 100 equal groups

Percentiles are symbolized by P1,P2, P3, ...P100

Dividing the distribution into 100 groups.

P1

1%

P2

1%

P99

1%

Percentiles

The percentile corresponding to a given value X is found using the following formula:

Percentile = (number of values below X) + .5

total number of values100%

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

18,15,12,6,8,2,3,5,20,10

Now use the data to determine the value corresponding with the 25th percentile.

Formula to use: c = n*p100

n = # valuesp = percentile

if c is not a whole number: round up to the nearest whole numberif c is a whole number find the value halfway between the cth term and the c+1 term

Find the value that corresponds to the 60th percentile

2,3,5,6,8,10,12,15,18,20

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

89.5-104.5

24

104.5-119.5

62

119.5-134.5

72

134.5-149.5

26

149.5-164.5

12

164.5-179.5

4

Class Boundaries

Cum

ulat

ive

Perc

enta

ges

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

smallest data value

largest data valueQ

1Q2

Q3

MD

25% 25% 25% 25%

Q1

Q2 --> The Median!

Q3

Finding the Quartiles

--> The median of the data below Q2

--> The median of the data above Q2

Find Q1,Q2 and Q3 for the following data set.

15,13,6,5,12,50,22,18

Deciles

Deciles divide the data into ______ groups.

We can use the formula for Percentiles to find Deciles

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

OutlierAn extremely high or extremely low value when compared to the rest of the data.

Outliers affect:- mean-standard deviation-range

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 IQR3. Multiply the IQR by 1.54. 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.

Check the following for outliers5,6,12,13,15,18,22

Using the Calculator

1. Enter data into L12. Press stat3. Move the arrow 1 right to Calc4. Press 1 for Var-Stats5. Press 2nd L1 then enter

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 valuesSx: sample standard deviation : population standard deviationminX: smallest data valueQ1: lower quartileMed: medianQ3: upper quartilemaxX: largest data value

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

Using the Calculator

For grouped data...1. Enter midpoints into L12. Enter frequencies into L23. Press stat button4. Use arrow to move 1 right to calc5. Press 1 for vars stats6. Press 2nd L1 and 2nd L2 then enter

Using the Calculator

Find the mean and standard deviation of the following data.

5.5-10.5 1 810.5-15.5

2 13

15.5-20.5

3 18

20.5-25.5

5 23

25.5-30.5

4 28

30.5-35.5

3 33

35.5-40.5

2 38

Using the Calculator

Graph a percentile graph on your calculator

5.5-10.5 110.5-15.5

2

15.5-20.5

3

20.5-25.5

5

25.5-30.5

4

30.5-35.5

3

35.5-40.5

2