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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=
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. 2
b. -1
c. 0
d. -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%
1%
P
2
P9
9
1%
Percentiles
The percentile corresponding to a given value X is found
using the following formula:
(number of values below X) + .5
Percentile =
total number of values
100%
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.
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
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
Frequenc cumulativ cumulativ
y
e
e percent
frequency
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
Cumulative Percentages
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
MD
smallest data
value
Q
1
25%
Q
2
25%
largest data
value
Q
3
25%
25%
Finding the Quartiles
Q2 --> The Median!
Q
1
--> The median of the data below Q2
Q
3
--> 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
Outlier
An extremely high or extremely low value when compared
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 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.
Check the following for outliers
5,6,12,13,15,18,22
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
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
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 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
Using the Calculator
Find the mean and standard deviation of the
following data.
5.5-10.5
10.5-15.5
15.5-20.5
20.5-25.5
25.5-30.5
30.5-35.5
35.5-40.5
1
2
3
5
4
3
2
8
13
18
23
28
33
38
Using the Calculator
Graph a percentile graph on your calculator
5.5-10.5
10.5-15.5
15.5-20.5
20.5-25.5
25.5-30.5
30.5-35.5
35.5-40.5
1
2
3
5
4
3
2