3.3 Measures of Position
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3.3 Measures of Position
Measures of location in comparison to the mean.- standard scores- percentiles- deciles- quartiles
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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.
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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
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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.
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Test 1: X = 38, X = 40 and s = 5
Test 2: X = 94, X = 100, s = 10
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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
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Percentiles
Percentiles divide the data set into 100 equal groups
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Percentiles are symbolized by P1,P2, P3, ...P100
Dividing the distribution into 100 groups.
P1
1%
P2
1%
P99
1%
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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%
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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
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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
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Find the value that corresponds to the 60th percentile
2,3,5,6,8,10,12,15,18,20
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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
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Class Boundaries
Cum
ulat
ive
Perc
enta
ges
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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
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smallest data value
largest data valueQ
1Q2
Q3
MD
25% 25% 25% 25%
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Q1
Q2 --> The Median!
Q3
Finding the Quartiles
--> The median of the data below Q2
--> The median of the data above Q2
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Find Q1,Q2 and Q3 for the following data set.
15,13,6,5,12,50,22,18
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Deciles
Deciles divide the data into ______ groups.
We can use the formula for Percentiles to find Deciles
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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
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OutlierAn extremely high or extremely low value when compared to the rest of the data.
Outliers affect:- mean-standard deviation-range
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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.
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Check the following for outliers5,6,12,13,15,18,22
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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
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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
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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
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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
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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
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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
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