Download z scores - Plainfield Public Schools

z scores
March 13, 2014
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 scores
March 13, 2014
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
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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. z scores
March 13, 2014
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
z scores
March 13, 2014
Percentiles
Percentiles divide the data set into 100 equal groups z scores
March 13, 2014
Percentiles are symbolized by P1,P2, P3, ...P100
Dividing the distribution into 100 groups.
P99
P1 P2
1% 1%
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%
z scores
March 13, 2014
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*p
100
n = # values
p = percentile
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
z scores
March 13, 2014
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
89.5­104.5
24
104.5­119.5
62
119.5­134.5
72
134.5­149.5
26
149.5­164.5
12
164.5­179.5
4
cumulative cumulative frequency percent
March 13, 2014
Cumulative Percentages
z scores
Class Boundaries
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
z scores
March 13, 2014
MD
smallest data value
Q1
25%
Q2
25%
largest data value
Q3
25%
25%
Finding the Quartiles
Q2 ­­> The Median!
Q1 ­­> The median of the data below Q2 Q3 ­­> The median of the data above Q2 z scores
March 13, 2014
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
z scores
March 13, 2014
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 to the rest of the data. Outliers affect:
­ mean
­standard deviation
­range
z scores
March 13, 2014
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
z scores
March 13, 2014
Using the Calculator
1. Enter data into L1
2. Press stat
3. Move the arrow 1 right to Calc
4. Press 1 for Var­Stats
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
z scores
March 13, 2014
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
z scores
March 13, 2014
Using the Calculator
Find the mean and standard deviation of the following data.
5.5­10.5
10.5­15.5
15.5­20.5
20.5­25.5
25.5­30.5
30.5­35.5
35.5­40.5
1
2
3
5
4
3
2
8
13
18
23
28
33
38
Using the Calculator
Graph a percentile graph on your calculator
5.5­10.5
10.5­15.5
15.5­20.5
20.5­25.5
25.5­30.5
30.5­35.5
35.5­40.5
1
2
3
5
4
3
2
z scores
March 13, 2014