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STANDARD DEVIATION
*Standard deviation can typically be described as the deviation a piece of date differs
from the mean.
*Standard deviation can almost be described as the “mean of the mean” it is a form of
statistical analysis
*You do this when you have data that is distributed normally or Normal distribution.
This means that the pieces of data don’t differ from the mean in extremes.
*Standard deviation uses past history or data to extrapolate future events. It shows the
tendency of the data and how it is spread out.
(--Subtracted from the mean--+-----added to the mean------)
*This is known as a bell curve. The line through the middle represents the mean or
average of the data. Each section is a percent of the data and is one standard deviation
away from the mean
Red- the red makes up 68% of the entire data; each half is only 34%. (Most of the data
falls in this category)
Green- each half of the green is 14% making the green a total of 28% but added to the red
makes up for approximately 95% of the data.
Blue- Blue represent 2% each but added to the total amount it is 99% (almost all data in
this section would be outliers)
White- a very small percent often not recognized as part of the data (anything in this
section would be an extreme outlier)
Fraction
Number of Standard
Of Data
50.0%
68.3
90.0
95.0
95.4
98.0
99.0
99.7
Deviations from Mean
.674
1.000
1.645
1.960
2.000
2.326
2.576
3.000
*This shows the percent of the data will fall in how many standard deviations
*Most of the data is found within the first two standard deviations away from the mean
negative or positive
*The wider and lower bell curves represent a higher standard deviation amount
*A steeper more condensed curve represents a smaller standard deviation amount and the
data is more clustered around the mean
*One form of calculating an average is the mean.
*The mean is the sum of all the pieces of data divided by the total number of pieces of
data
Mean = Sum of data
Total # pieces of data
Ex. 5,6,7,5,6,4,8,9
Sum of data
Total # pieces of data
50
8
Mean = 6.25
*One equation to determine the standard deviation is
S= Standard Deviation
= The sum of the pieces of data
X= your piece of data
M= mean
N= your number of pieces of data
*However all this seems very complicating to find of standard deviation when there is an
easier way to do it
*You will need several pieces of data to determine the mean
Ex. 5,7,4,9,5,7,6,4,8,5
So now you need to calculate the mean
Mean = Sum of data
Total # pieces of data
60
10
Mean= 6
Now you need to subtract each piece of data from the mean and then square it

Even if you are subtracting a larger number than the mean and it becomes
negative don’t worry because when you square a negative it becomes positive.
#
Data piece
Mean Subtracted From Data
Squared Answer
1
2
5
7
5-6= -1
7-6= 1
1
1
3
4
5
4
9
5
4-6= -2
9-6= 3
5-6= -1
4
9
1
6
7
8
7
6
4
7-6= 1
6-6= 0
4-6= -2
1
0
4
9
10
8
5
8-6= 2
5-6= -1
4
1
Total= 26
*Now you take the total number of squared answers and divide it by the number of pieces
of data
Total Squared Ans
# Of pieces of data
26
10
=
2.6
*The total squared answer can also be called the Variance
*Now take the square root of the number and you have your Standard deviation
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (2.6)
Standard Deviation = 1.61
*There you have your deviation number!
*This is a graph of a bell curve and how it would be labeled after you find the specific
standard deviation
1.17------2.78------4.39------- 6 --------7.61------9.22-----10.83
*You can determine the percentage error
Percentage error=(Standard Deviation / Mean) x 100
Percentage error= (1.61 / 6) x 100
Percentage error= (.268) x 100
Percentage error= 26.8%
*This is a good indicator on whether or not the mean is a good statistic
*Standard deviation gives you a basis for interpreting the data in regards to probability
*The Variance is the total of the pieces of data subtracted from the mean then squared
Variance= (∑Data piece – Mean) 2
*Standard Deviation is typically used as a component with other indicators to give a real
statistical analysis of the given data
*Standard Deviation can be used with a variety of real life situations like, stock market
analysis, medical concepts, showing the differences in various pieces of data
*But typically it is used to show stock market change and analysis.
Ex. Lets take 20 samples of stock market closing to determine the mean and standard
deviation of the stock
#
1
2
3
4
Piece of data
109.00
103.06
102.75
108.00
Mean – Piece of data
109.00-112.30=-3.30
103.06-112.30=-9.24
102.75-112.30=-9.55
108.00-112.30=-4.30
Squared Answer
10.91
85.38
91.26
18.52
5
107.56
107.56-112.30=-4.74
22.47
6
7
8
9
105.25
107.69
108.63
107.00
105.25-112.30=-7.05
107.69-112.30=-4.62
108.63-112.30=-3.68
107.00-112.30=-5.30
49.75
21.30
13.53
28.12
10
11
12
109.00
110.00
112.75
109.00-112.30=-3.30
110.00-112.30=-2.30
112.75-112.30=0.45
10.91
5.30
0.20
13
14
15
113.50
114.25
115.25
113.50-112.30=1.20
114.25-112.30=1.95
115.25-112.30=2.95
1.43
3.79
8.68
16
17
121.50
126.88
121.50-112.30=9.20
126.88-112.30=14.57
84.58
212.34
18
19
122.50
119.00
122.50-112.30=10.20
119.00-112.30=6.70
103.97
44.85
20
122.50
122.50-112.30=10.20
103.97
Total= 921.28
#1. Mean = Sum of data
Total # pieces of data
2246.06
20
Mean= 112.30
#2. Total Squared Ans
# Of pieces of data
921.28
20
=
46.064
#3. Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (46.064)
Standard Deviation = 6.787
Percentage error=(Standard Deviation / Mean) x 100
Percentage error = (6.787 / 112.30) x 100
Percentage error = (.060400) x 100
Percentage error = 6.04%
*This
is the graph of the IBM stock that we took the information from to form our
standard deviation to show how standard deviation isn’t always the best way to represent
the statistics of a particular set of data
*Standard deviation is frequently used as a measure of the volatility of a random financial
variable
*Here are some graphs of financial volatility
*You can use the mean or median when trying to calculate the standard deviation if the
data is normally distributed because they should both be very close
*The Median is the middle number or term in a series of numbers that are in order from
least to greatest
Ex. 2,2,2,3,3,3,4,4,5,6,6,7,7
The median is 4
The mean would be
Mean = Sum of data
Total # pieces of data
54
13
Mean= 4.15
So the mean and median in this case are very similar
Questions
1) Find the mean and standard deviation for this set of data.
97, 100, 99, 95, 97, 101, 99
#
Data
piece
Data piece - Mean
Squared
Answer
1
2
3
4
5
6
7
Total=
2. Find the mean for the given data:
0.60, 0.94, 0.76, 1.02, 0.68, 0.72, 0.79, 0.92
Mean =
3. Mrs. Wells gave a math test to 12 random students the scores were as follows:
79, 84, 92, 78, 76, 85, 94, 89, 77, 81, 83, 96
Find the mean and standard deviation for the test scores
#
1
2
3
Data Piece
Data piece - mean
Squared answer
4
5
6
7
8
9
10
11
12
Total =
4. A grade 3 test was given to 1000 randomly selected grade 10 students would you
expect the results to be normally distributed, explain why or why not.
5.
Class
1
Mean sleeping times
8h
Standard deviation
1.2h
2
3
10h
9h
2h
0.9h
4
7h
1.5h
A) A student slept for 12h, which class would you expect he is most likely from?
B) Another student slept for 8h why is it hard to say which class he was form?
6. A restaurant seats 162 people, average stay was 56 minutes, the standard deviation is 8
minutes. Would be unusual for some one to stay for one hour and 5 minute?
7. These are the earnings of 9 families in a neighborhood:
27 000, 29 000, 25 000, 32 000, 30 500, 28 500, 35 000, 27 500, 31 000
Find the standard deviation, mean, percentage error and label the bell curve
#
Data Piece
Data Piece - Mean
Squared Answer
1
2
3
4
5
6
7
8
9
Total=
8. On average students get 9 hours of sleep per night, the standard deviation is 1.2
hours, fill in the blanks
A) 68% of the students slept between ______h and ______h
B) 95% of the students slept between ______h and ______h
9. A sample of 10 silver maple’s height was conducted and they found the heights
were as follows: 2.4m, 2m, 3.2m, 2.6m, 3.1m, 1.9m, 2.2m, 3.6m, 3.5m, and 2.1m
Find the standard deviation, mean and percentage error
#
1
2
Data Piece
Data piece - Mean
Squared Answer
3
4
5
6
7
8
9
10
Total=
10. If the average return from an investment of 1000 dollars is 275 dollars and the
standard deviation is 15.5. How likely is it to get a return of 315 dollars? How
many standard deviations away from the mean would it be?
11. A variance of 39.24 was found and there were 9 pieces of data find the standard
deviation
12. Calculate the median of the following data:
13,16,18,14,18,12,17,13,15,19,13,15,14
13. On average students spend 4 hours on math homework per day. The standard
deviation is 0.35
A) What percent of students do math homework between 3.65 to 4.35 hours
per night?
B) What percent of math students do math homework between 3.3 to 4.7
hours per night
14. If you have a variance of 67 and the total number of pieces of data are 25 what is
your standard deviation?
15. The average student tutorial kit was 20 pages with a standard deviation of 1.5
A) 68% of the student’s tutorial kits were from _____ to _____ pages.
B) 95% of the student’s tutorial kits were from _____ to _____ pages.
Appendix
#1
Mean = Sum of data
Total # pieces of data
688
7
Mean = 98.2
#
Data piece - Mean
1
Data
piece
97
97 - 98.2= -1.2
Squared
Answer
1.44
2
3
4
100
99
95
100 - 98.2= 1.8
99 - 98.2= 0.8
95 - 98.2= -3.2
3.24
0.64
10.24
5
6
7
97
101
99
97 – 98.2= -1.12
101 – 98.2= 2.8
99 – 98.2= 0.8
1.44
7.84
0.64
Total= 25.48
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (3.64)
Standard Deviation = 1.9
#2
Mean = Sum of data
Total # pieces of data
6.43
8
Mean= 0.80
#3
Mean = Sum of data
Total # pieces of data
1014
12
Mean= 84.5
#
Data Piece
Data piece - mean
Squared answer
1
79
79 – 84.5= -5.5
30.25
2
3
4
84
92
78
84 – 84.5= -0.5
92 – 84.5= 7.5
78 – 84.5= -6.5
0.25
56.25
42.45
5
6
7
76
85
94
76 – 84.5= -8.5
85 – 84.5= 0.5
94 – 84.5= 9.5
72.25
.25
90.25
8
9
10
11
12
89
77
81
83
96
89 – 84.5= 4.5
77 – 84.5= -7.5
81 – 84.5= -3.5
83 – 84.5= -1.5
96 – 84.5= 11.5
20.25
56.25
12.25
2.25
132.25
Total = 515.2
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (42.93)
Standard Deviation = 6.55
#4
The data would not be evenly distributed because the data would be clustered around the
100% mark because it would be very easy to grade 10 students.
#5
5a) The student would most likely be from class two because their average sleeping time
is the highest
5b) It is hard to tell because class one three and four and be in the first standard deviation
away
#6
It is not very unusual
#7
Mean = Sum of data
Total # pieces of data
265 500
9
Mean= 29 500
#
Data Piece
Data Piece - Mean
Squared Answer
1
27 000
27 000 - 29 500= -2 500
6 250 000
2
3
4
5
29 000
25 000
32 000
30 500
29 000 - 29 500= -500
25 000 - 29 500= -4 500
32 000 - 29 500= 2 500
30 500 - 29 500= 1 000
250 000
20 250 000
6 250 000
1 000 000
6
7
8
9
28 500
35 000
27 500
31 000
28 500 - 29 500= -1 000
35 000 - 29 500= 5 500
27 500 - 29 500= -1 500
31 000 - 29 500= 1 500
1 000 000
30 250 000
2 250 000
2 250 000
Total=69 750 000
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (7 750 000)
Standard Deviation = 2783.9
Percentage error=(Standard Deviation / Mean) x 100
Percentage error= (2783.9 / 29 500) x 100
Percentage error= (.0943) x 100
Percentage error= 9.43%
21148.3---------23932.2-------26716.1--------29500----------32283.9--------35067.8-------37851.7
#8
C) 68% of the students slept between 7.8h and 10.2h
D) 95% of the students slept between _6.6h and _11.4_h
#9
Mean = Sum of data
Total # pieces of data
26.6
10
Mean= 2.66
#
1
2
Data Piece
2.4
2.0
Data piece - Mean
2.4 – 2.66= -0.26
2.0 – 2.66= -0.66
Squared Answer
0.0676
0.435
3
4
5
3.2
2.6
3.1
3.2 – 2.66= 0.54
2.6 – 2.66= -0.06
3.1 – 2.66= 0.44
0.291
0.0036
0.193
6
7
8
1.9
2.2
3.6
1.9 – 2.66= -0.76
2.2 – 2.66=-0.46
3.6 – 2.66= -0.94
0.557
0.211
0.883
9
10
3.5
2.1
3.5 – 2.66= 0.84
2.1 – 2.66= -0.56
0.705
0.313
Total= 3.61
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (0.361)
Standard Deviation = 0.6
Percentage error=(Standard Deviation / Mean) x 100
Percentage error= (0.6 \ 2.66) x 100
Percentage error= (0.225) x 100
Percentage error= 22.5%
#10
It is not very likely due to the fact that it is three standard deviations away
#11
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (4.36)
Standard Deviation = 2.08
#12
Unsorted data
13,16,18,14,18,12,17,13,15,19,13,15,14
Sorted Data
12, 13, 13, 13, 14, 14, 15, 15, 16, 17, 18, 18, 19
Median= 15
#13
A) 68%
B) 95%
#14
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (2.68)
Standard Deviation = 1.64
#15
A) 68% of the student’s tutorial kits were from __18.5_ to _21.5__ pages.
B) 95% of the student’s tutorial kits were from 17__ to __23_ pages.
Quiz
1) 960 students took a quiz marked out of 175. The results were normally distributed and
the mean was equal to 115 the standard deviation was 25; within what range would 68%
of the students fall under?
2) Someone claimed they got 170 on the same test. Would you believe them? Why or
why not?
3) The standard deviation is 2 and the mean is 6 label the given graph including the mean
and each standard deviation
4) What does this mean ()?
5) What is the difference between mean and median?
6) The grades of all the boys in a math class was recorded and the results were as follows:
79, 88, 92, 86, 85, 82, 76, 87, 90, 83
Find the mean and standard deviation
#
1
2
3
4
Data Piece
Data Piece - Mean
Squared Answer
5
6
7
8
9
10
Total=
7) What is the definition of variance?
8) 10 runners each ran a 200-meter sprint, they all had different times they are:
30s, 29s, 34s, 33s, 31s, 28s, 30, 32s, 26s, 27s
Find the Mean, standard deviation, percentage error and label the deviations on the graph
provided
#
1
2
Data piece
Data Piece - Mean
Squared Answer
3
4
5
6
7
8
9
10
Total=
Quiz Answers
#1
90 to 140
#2
It would be improbable but not impossible
#3
0-----------2----------4----------6----------8----------10--------12
#4
The sum of all the pieces of data
#5
The mean is the sum of all the numbers / by number of pieces of data
Or
Mean = Sum of data
Total # pieces of data
The median is the middle number in a series of numbers in ascending or descending order
#6
Mean = Sum of data
Total # pieces of data
848
10
Mean= 84.8
#
1
2
3
4
Data Piece
79
88
92
86
Data Piece - Mean
79 - 84.8= -5.8
88 - 84.8= 3.2
92 - 84.8= 7.2
86 - 84.8=1.2
Squared Answer
33.64
10.24
51.84
1.44
5
6
7
85
82
76
85 - 84.8= 0.2
82 - 84.8=-2.8
76 - 84.8=-8.8
0.04
7.84
77.44
8
9
10
87
90
83
87 - 84.8=2.2
90 - 84.8= 5.2
83 - 84.8= -1.8
4.84
27.04
3.24
Total= 217.6
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (21.76)
Standard Deviation = 4.66
#7
The variance is the total number of squared numbers, so the data piece minus the mean
squared
Or
Variance= (total number of squared numbers – Mean) 2
#8
Mean = Sum of data
Total # pieces of data
300
10
30
30s, 29s, 34s, 33s, 31s, 28s, 30, 32s, 26s, 27s
#
1
2
Data piece
30
29
Data Piece - Mean
30 – 30= 0
29 – 30= -1
Squared Answer
0
1
3
4
5
34
33
31
34 – 30= 4
33 – 30= 3
31 – 30= 1
16
9
1
6
7
8
28
30
32
28 – 30= -2
30 – 30= 0
32 – 30= 2
4
0
4
9
10
26
27
26 – 30= -4
27 – 30= -3
16
9
Total= 60
Standard Deviation = √ (of squared total / # of pieces of data)
Standard Deviation = √ (6)
Standard Deviation = 2.44
Percentage error=(Standard Deviation / Mean) x 100
Percentage error= (2.44 \ 30) x 100
Percentage error= (.081) x 100
Percentage error= 8.1%
22.68---------25.12--------27.56----------30------------32.44-------34.88-------37.32
Bibliography
http://bmj.com/collections/statsbk/2.shtml
http://www.contingencyanalysis.com/glossarystandarddeviation.ht
m
http://www.stockcharts.com/education/What/IndicatorAnalysis/ind
ic_standardDev.html
http://www.trade10.com/standard_deviati.htm
http://www.crpc.rice.edu/CRPC/GT/bchristo/lessons/StanDev1.ht
ml
http://www.ruf.rice.edu/~lane/hyperstat/A16252.html