A Interval data: Standard deviation (abbreviated
S when used for a sample)
1. ungrouped data: S = Where: x = variable
N = total # in sample
(Caution: x^{2} means to square
all the x's, then sum them up)
Example:
# of siblings | x^{2} |
0 | 0 |
0 | 0 |
0 | 0 |
0 | 0 |
1 | 1 |
1 | 1 |
1 | 1 |
2 | 4 |
2 | 4 |
2 | 4 |
3 | 9 |
4 | 16 |
16 = x | 40= x^{2} |
Q: What is the spread or variation in number of siblings?
A: s =
s = =
s = .0833(14.97) = 1.25
2. grouped data: s
=
Where: i = width of category
f = frequency
N = total cases in sample (or f )
d = step deviations from an arbitrary starting point
* In this class, all "d's" will appear
the same.
Example:
# of siblings | f | d | fd | d^{2} | fd^{2} |
0 | 4 | 0 | 0 | 0 | 0 |
1 | 3 | 1 | 3 | 1 | 3 |
2 | 3 | 2 | 6 | 4 | 12 |
3 | 1 | 3 | 3 | 9 | 9 |
4 | 1 | 4 | 4 | 16 | 16 |
i = 1 | 12 = N | 16 = fd | 40 = fd^{2} |
Q: What is the spread or variation in the number of siblings?
A:
S =
S =1.25
STANDARDIZED SCORES: Z-SCORES =
# of siblings | f | fm | |||
0 | 4 | 0 | 0 - 1.333 | -1.333/1.25 | -1.07 |
1 | 3 | 3 | 1 - 1.333 | -.333/1.25 | -0.27 |
2 | 3 | 6 | 2 - 1.333 | .667/1.25 | 0.53 |
3 | 1 | 3 | 3 - 1.333 | 1.667/1.25 | 1.33 |
4 | 1 | 4 | 4 - 1.333 | 2.667/1.25 | 2.13 |
12 | 16 |
* This example assumes normal distribution
even though it is not normally distributed.
Z-SCORE CHART (selected Z-scores)
Z-Scores | Area between the mean and the score | Area beyond the mean and the score |
.27 | 0.1064 | 0.3939 |
.53 | 0.2019 | 0.2981 |
1.00 | 0.3413 | 0.1587 |
1.07 | 0.3577 | 0.1423 |
1.33 | 0.4082 | 0.0918 |
2.00 | 0.4772 | 0.228 |
2.13 | 0.4834 | 0.0166 |
3.00 | 0.4986 | 0.0014 |
NORMAL DISTRIBUTION
+ 1 s will include 68.26% of the cases
+ 2 s will include 95.44% of the cases
+ 3 s will include 99.72%
of the cases
so plus or minus 1 SD = 68.26% of the population and 2SD = 95.44% and 3SD = 99.72%
Example: Bank balance (Mr. Jones):
x = $2000.
s = 500 and distribution = normal
Interpretation:
68% of the time, Mr. Jones's balance is between $1500 - $2500.
95% of the time, " " " " $1000 - $3000.
99.7% of the time, " " " " $500
- $3500.
Example: Intelligence Quotient Scores
x = 100 S = 16
Interpretation:
68% of the sample's IQs fall between 86 and 116.
95% of the sample's IQs fall between 70 and 132
99.7% of the sample's IQs fall between 54 and 148
SMOOTH, UNIMODAL (BUT SKEWED) DISTRIBUTION
+ 1s will include 56% of the cases, or more.
+ 2s will include 89% of the cases, or more.
+ 3s will include 95%
of the cases, or more.
Example: Class Averages from SOC201
x = 72.76%
S = 15.48
Interpretation:
56% or more of the class will average between 57.28 and 88.24%.
89% or more of the class will average between 41.8 and 103.72%
95% or more of the class will average between 26.32 and 119.2%
* It is impossible to average more than 110%,
in most classes 100%. In such cases, reword the statements to
reflect this, i.e. "26.32 and 110%"
ANY SHAPED DISTRIBUTION
(irregular): use this if the distribution of the actual raw data
is completely unknown. Notice it has no interpretation for +
1s.
+ 2s will include 75% of the cases, or more.
+ 3s will include 89% of the cases, or more.
Example: Hours Soc201 students are employed per week
x = 21.45 Distribution irregular
S = 9.63
Interpretation:
75% or more of the students are employed between 2.2 and 40.7 hrs. a week.
89% or more of the students are employed between 0 and 50.33 hours a week.*
*It is impossible to be employed less than
0 hours per week.
Example: GPA's in Soc201 classes
x = 2.7
S = .4 distribution = irregular
Interpretation:
75% or more of the Soc201 students have GPA's between 1.9 and 3.5
89% or more of the Soc201 students have GPA's between 1.5 and 3.9
Example: Frequency Distribution
# of siblings | f |
0 | 4 |
1 | 3 |
2 | 3 |
3 | 1 |
4 | 1 |
12 = N |
Q: S = 1.25 in the above frequency distribution, what does this mean?
A: 1) compute the mean: x = 1.33
2) identify the shape of distribution (make a
frequency polygon)
3) interpret the data
Interpretation:
75% or more of the sample have between 0 and 3.83 siblings.*
89% or more of the sample have between 0 and 5.08 siblings.*
*Impossible to have less than 0 siblings