measures of position
DESCRIPTION
mathematicsTRANSCRIPT
Measures of Position
Z Score
• Also called the standard score
Z Score
• Also called the standard score
• Represents the number of standard deviations a score is from the mean
Z Score
• Also called the standard score
• Represents the number of standard deviations a score is from the mean
• Always round value to 2 decimal places.
Formulas
• Sample
• Population
x - xz = s
x - z = µσ
Example
Human body temperatures have a mean of 98.20 degrees and a standard deviation of 0.62 degrees.
Find the z score for temperatures of: a. 100 degrees b. 97 degrees
Solution
Z = (100 – 98.20)/0.62 Z = 2.90
Solution
Z = (100 – 98.20)/0.62 Z = 2.90 Z = (97 – 98.20)/0.62 Z = -1.94
Significance of Z
• Z scores above 2 or below -2 are considered to be UNUSUAL.
• Z scores above 3 or below -3 are considered to be VERY UNUSUAL.
Conclusion about temperatures
• The temperature of 100 degrees is UNUSUAL.
• The temperature of 97 degrees is ordinary.
Another use of z scores
Z scores can also be used to compare
relative position for different data sets.
Example: page 100 #10
Example: page 100 #10
a. Z = (144 – 128)/34 = 0.47 b. Z = (90 – 86)/18 = 0.22 c. Z = (18 – 15)/5 = 0.60
The third score is the largest, so that is the
test result with the highest relative score.
Percentiles
• A percentile tells the percent of scores that are lower than a given score.
Percentiles
• A percentile tells the percent of scores that are lower than a given score.
• Write: P90 (or whatever number we need)
Percentiles
• A percentile tells the percent of scores that are lower than a given score.
• Write: P90 (or whatever number we need)
• We will not be calculating percentiles as
the data sets should be quite large in order for the percentile to be meaningful.
Example
• A pediatrician reports that a child is in the 90th percentile for heights among children of that age. This is P90.
• That means 90% of all children of that age are shorter than the given child. The child is taller than average.
Quartiles
• Quartiles divide the data set into 4 groups, each of which has the same number of members.
• Q1 corresponds to P25 • Q2 corresponds to P50 or the median • Q3 corresponds to P75
Q1, Q2, Q3 divides ranked scores into four equal parts
Quartiles
25% 25% 25% 25%
Q3 Q2 Q1 (minimum) (maximum)
(median)
Finding quartiles
1. Sort the data.
Finding quartiles
1. Sort the data. 2. Locate the median.
Finding quartiles
1. Sort the data. 2. Locate the median. 3. Q1 is the median of the group of scores
starting at the minimum value and going up to but not including the true median.
Finding quartiles
1. Sort the data. 2. Locate the median. 3. Q1 is the median of the group of scores
starting at the minimum value and going up to but not including the true median.
4. Q3 is the median of the group of scores starting just past the true median and going up to the maximum value.
Example
• Use Harry Potter data found on page 69 # 2
70.9 74.0 78.6 79.2 79.5 80.2 82.5 83.7 84.3 84.6 85.3 86.2
Median
• The median is the average of the 6th and 7th scores.
• (80.2+ 82.5)/2 • 81.35
70.9 74.0 78.6 79.2 79.5 80.2 82.5 83.7 84.3 84.6 85.3 86.2
Q1
• Find the median of the first 6 scores
• (78.6 + 79.2)/2 • 78.9
70.9
74.0
78.6
79.2
79.5
80.2
Q3
• Find the median of the last 6 scores
• (84.3+84.6)/2 • 84.45
82.5
83.7
84.3
84.6
85.3
86.2
Another Example
Weights of regular coke
0.8150
0.8163
0.8181
0.8192
0.8211
0.8247
Another Example
Weights of regular coke Median
(0.8181+0.8192)/2 0.81865
0.8150
0.8163
0.8181
0.8192
0.8211
0.8247
Another Example
Weights of regular coke Median
(0.8181+0.8192)/2 0.81865 Q1 0.8163
0.8150
0.8163
0.8181
Another Example
Weights of regular coke Median
(0.8181+0.8192)/2 0.81865 Q1 0.8163 Q3 0.8211
0.8192
0.8211
0.8247
Using the TI
• We can check results with the TI calculator.
• Put the data into a list. • Press STAT, CALC, One-Var stats • Enter the name of the list • Scroll down to see the values
Harry Potter results
• Here is the screen output: