s519: evaluation of information systems social statistics chapter 7: are your curves normal?
TRANSCRIPT
S519: Evaluation of Information Systems
Social Statistics
Chapter 7: Are your curves normal?
This week
Why understanding probability is important? What is normal curve How to compute and interpret z scores.
What is probability?
The chance of winning a lottery The chance to get a head on one flip of a
coin Determine the degree of confidence to state
a finding
Normal distribution
Figure 7.4 – P157 Almost 100% of the scores fall between (-3SD,
+3SD) Around 34% of the scores fall between (0, 1SD)
Are all distributions normal?
Normal distribution
The distance between contains Range (if mean=100, SD=10)
Mean and 1SD 34.13% of all cases 100-110
1SD and 2SD 13.59% of all cases 110-120
2SD and 3SD 2.15% of all cases 120-130
>3SD 0.13% of all cases >130
Mean and -1SD 34.13% of all cases 90-100
-1SD and -2SD 13.59% of all cases 80-90
-2SD and -3SD 2.15% of all cases 70-80
< -3SD 0.13% of all cases <70
Z score – standard score
If you want to compare individuals in different distributions
Z scores are comparable because they are standardized in units of standard deviations.
Z score
Standard score
X
z
X: the individual score
: the mean
: standard deviation
Sample or population?
Z scoreMean and SD for Z distribution?
Mean=25, SD=2, what is the z score for 23, 27, 30?
Z score
Z scores across different distributions are comparable
Z scores represent a distance of z score standard deviation from the mean
Raw score 12.8 (mean=12, SD=2) z=+0.4 Raw score 64 (mean=58, SD=15) z=+0.4
Equal distances from the mean
Comparing apples and oranges:
Eric competes in two track events: standing long jump and javelin. His long jump is 49 inches, and his javelin throw was 92 ft. He then measures all the other competitors in both events and calculates the mean and standard deviation:
Javelin: M = 86ft, s = 10ft Long Jump: M = 44, s = 4 Which event did Eric do best in?
Excel for z score
Standardize(x, mean, standard deviation) (x-average(array))/STDEV(array)
What z scores represent?
Raw scores below the mean has negative z scores
Raw scores above the mean has positive z scores
Representing the number of standard deviations from the mean
The more extreme the z score, the further it is from the mean,
What z scores represent?
84% of all the scores fall below a z score of +1 (why?)
16% of all the scores fall above a z score of +1 (why?)
This percentage represents the probability of a certain score occurring, or an event happening
If less than 5%, then this event is unlikely to happen
Exercise
In a normal distribution with a mean of 100 and a standard deviation of 10, what is the probability that any one score will be 110 or above?
What about 6σhttp://en.wikipedia.org/wiki/Six_Sigma
If z is not integer
Table B.1 (S-P357-358) NORMSDIST(z)
To compute the probability associated with a particular z score
Exercise
The probability associated with z=1.38 41.62% of all the cases in the distribution fall
between mean and 1.38 standard deviation, About 92% falls below a 1.38 standard deviation How and why?
Between two z scores
What is the probability to fall between z score of 1.5 and 2.5 Z=1.5, 43.32% Z=2.5, 49.38% So around 6% of the all the cases of the
distribution fall between 1.5 and 2.5 standard deviation.
Exercise
What is the percentage for data to fall between 110 and 125 with the distribution of mean=100 and SD=10
Exercise
The probability of a particular score occurring between a z score of +1 and a z score of +2.5
Exercise
Compute the z scores where mean=50 and the standard deviation =5 55 50 60 57.5 46
Exercise
The math section of the SAT has a μ = 500 and σ = 100. If you selected a person at random: a) What is the probability he would have a score
greater than 650? b) What is the probability he would have a score
between 400 and 500? c) What is the probability he would have a score
between 630 and 700?
Determine sample size
Expected response rate: obtain based on historical data
Number of responses needed: use formula to calculate
Rate Response Expected
Needed Responses ofNumber Size Sample
Number of responses needed
n=number of responses needed (sample size) Z=the number of standard deviations that
describe the precision of the results e=accuracy or the error of the results =variance of the data for large population size
2
22
e
Zn x
2
x
Deciding
from previous surveys intentionally use a large number conservative estimation
e.g. a 10-point scale; assume that responses will be found across the entire 10-point scale
3 to the left/right of the mean describe virtually the entire area of the normal distribution curve
=10/6=1.67; =2.78
2
x
2
Example
Z=1.96 (usually rounded as 2) =2.78 e=0.2 n=278 (responses needed) assume response rate is 0.4 Sample size=278/0.4=695
2
22
e
Zn x
2
Exercise
Z=1.96 (usually rounded as 2) 5-point scale (suppose most of the responses
are distributed from 1-4) error tolerance=0.4 assume response rate is 0.6 What is sample size?
2
22
e
Zn x