1. 3-3: Measures of Position
Instructor: Alaa saud
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2. 3-3: Measures of Position
1- Standard Scores or Z-score Quartiles Outliers.
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3. 1-Standard Score (Z score)
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4. 1-Standard Score (Z score)
A Z Score or Standard Score for a value is obtained by subtracting the mean from the value and dividing the result by Standard deviation . The symbol for a standard score is (Z).
The z score represents the number of standard deviations that a data value falls above or below the mean.
The formula is
For Sample
For Population

5. Note that :
If the Z score is positive, the score is above the mean.
If the Z score =0,the score is the same as the mean.
If the Z score is negative,the score is below the mean

6. Compare her relative position on the two tests.? Solution :
Example 3-29
A student score 65 on a calculus test had a mean of 50 and a standard deviation of 10 ;she scored 30 on a history test with a mean of 25 and standard deviation of 5.
Compare her relative position on the two tests.?
Solution :
For calculus :
For history:
Since the Z score for calculus is larger ,her relative position in the calculus higher than in history .
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7. Find the z score for each test ,and state which is higher .
Example 3-30:
Find the z score for each test ,and state which is higher .
Solution:
For test A,
For test B ,
S=5
X=40
X=38
Test A
S=10
X=100
X=94
Test B
The score for test A is relatively higher than the score for the test B
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8. 2- Quartiles
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9. 2- Quartiles
Quartiles divide the distribution into groups, separated by
Largest
value
Smallest
value Q1 Q2 Q3
25%
50%
75%
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10. The Procedure: To Find Q1,Q2,Q3:
Step1: Arrange the data in order from lowest to highest .
Step2:Find the median of the data values . This value is Q2.
Step3:Find the median of the data values that fall below Q2,this is the value for Q1.
Step4:Find the median of the data values that fall above
Q2,this is the value for Q3.
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11. Example 3-36: Find Q1,Q2 ,Q3 for the data set 15,13,6,5,12,50,22,18.
Solution:
Step1: Arrange data in order from lowest to highest.
5,6,12,13,15,18,22,50
Step2:find the median Q2
Step3: find the median of the data value less than 14
5,6,12,13
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12. Step4:find the median of the data values greater than 14
15,18,22,50
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13. 3-Outliers
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14. Step 1: Arrange the data in order and find Q1 and Q3.
An Outlier is an extremely high or an extremely low data value when compared with the rest of the data values .
Step 1: Arrange the data in order and find Q1 and Q3.
Step 2: Find the interquartile range IQR= Q3 - Q1
Step 3: Multiply the IQR by 1.5 .
Step 4: Subtract the value obtained in step 3 form Q1 and add the value to Q3.
Step 5: Check the data set for any data value that is smaller than Q1-1.5(IQR) or larger than Q3+1.5(IQR).
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15. Check the following data set for outliers 5,6,12,13,15,18,22,50
Example 3-37:
Check the following data set for outliers ,6,12,13,15,18,22,50
Solution:
Step1: find Q1,Q2,Q3
Step2 :find interquartile range(IQR)
Step3:Multiply this value by 1.5
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16. check the data value that fall outside the interval [-7.5 ,36]
Step4: subtract the value obtained in step3 from Q1, and add the value obtained in step3 to Q3
and
Step5:
check the data value that fall outside the interval
[-7.5 ,36]
the value 50 is outside this interval , so 50 is an outlier
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17. 3-4: Exploratory Data Analysis
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18. The five –Number Summary : 1-lowest value of the data set . 2-Q1.
3-the median Q2.
4-Q3.
5-the highest value of the data set .
A Boxplot can be used to graphically represent the data set .
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19. Procedure for constructing a boxplot
Find five -Number summary .
Draw a horizontal axis with a scale such that it includes the maximum and minimum data value .
Draw a box whose vertical sides go through Q1 and Q3,and draw a vertical line though the median .
Draw a line from the minimum data value to the left side of the box and line from the maximum data value to the right side of the box. Q1 Q2 Q3
maximum
minimum
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20. Compare the distributions using Box Plot s?
Example 3-39:
Cheese substitute
Real cheese
290
250
180
270 40 45
420
310
340
260
130 90
240
220
Compare the distributions using Box Plot s?
Solution :
Step1: find Q1,MD,Q3 for the Real cheese data
40 ,45 ,90 ,180 ,220 ,240 ,310 ,420 Q1 MD Q3

21. Step2:find Q1,Q2 and for the cheese substitute data
130 ,180 ,250 ,260 ,270 ,290 ,310 ,340 Q1 Q2 Q3
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22.
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23. Information obtained from a Box plot
The median(MD) is near the center.
The distribution is symmetric.
The median falls to left of the center .
The distribution is positively skewed (Left skewed).
The median falls to right of the center .
The distribution is negatively skewed (Right skewed).
The lines are the same length.
The distribution is symmetric.
The right line is larger than the left line .
The distribution is positively skewed (Right skewed). .
The left line is larger than the right line .
The distribution is negatively skewed (Left skewed).
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24. A student scored 75 on a STAT exam with a z-score equals (-1),then his score is considered to be :
Above the mean
Same as the mean
Below the mean
None of the above
If a student scored X points on a test where the mean score was 7.80, the standard deviation was 2.05 and the student's z-score was 1.07, then X must be 10
0.3
-10
-0.3

25. 3. If Q1=4, IQR=40, Find Q3: 10 44 40 -4
4. All the values in a dataset are between 60 and 88, except for one value of 97. That value 97 is likely to be
An outlier
The mean
The box plot
The range

26. 5. Calculate the appropriate measure of central tendency for the following data: 10 , 20 , 180 , 25 , 22 , 18 a) b) 21 c) 95 d) What is the value of the mode when all values in the data set are different? a) 0 b) 1 c) There is no mode. d) It cannot be determined unless the data values are given

27. The End
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