Download Statistics: Range, Standard Deviation

Statistics: Range, Standard Deviation
Contemporary Math
Josh Engwer
TTU
03 August 2015
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
1 / 23
Summarizing Data by Measuring its Dispersion
One way to summarize data is to visualize the data using:
Bar Graphs
Histrograms
Stem-and-Leaf Displays
One way to numerically summarize data is to measure its central tendency:
Mean
Median
Mode
A 2nd way to numerically summarize data is to measure its dispersion:
Range
Standard Deviation
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
2 / 23
Range & Standard Deviation of a Data Set (Definition)
Definition
(Range of a Data Set)
Given a data set with n data values.
Then the range is the difference between the largest and smallest values.
Definition
(Standard Deviation of a Data Set)
Given a data set with n data values.
Then the standard deviation measures ”how spread out” the data is:
s
P
2
(x − x)
s=
n−1
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
3 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
4 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
(a) Range = (Largest Data Value) − (Smallest Data Value)
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
5 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 15 − 3 = 12
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
6 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 15 − 3 = 12
P
x
(b) x =
n
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
7 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 15 − 3 = 12
P
x
3 + 15 + 8 + 11 + 15
52
(b) x =
=
=
= 10.4
n
5
5
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
8 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 15 − 3 = 12
P
x
3 + 15 + 8 + 11 + 15
52
(b) x =
=
=
= 10.4
n
5
5
s
P
2
(x − x)
(c) s =
n−1
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
9 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 15 − 3 = 12
P
x
3 + 15 + 8 + 11 + 15
52
=
=
= 10.4
(b) x =
n
5
5
s
P
2
(x − x)
(c) s =
n−1
r
(3 − 10.4)2 + (15 − 10.4)2 + (8 − 10.4)2 + (11 − 10.4)2 + (15 − 10.4)2
=
5−1
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
10 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 15 − 3 = 12
P
x
3 + 15 + 8 + 11 + 15
52
(b) x =
=
=
= 10.4
n
5
5
s
P
2
(x − x)
(c) s =
n−1
r
(3 − 10.4)2 + (15 − 10.4)2 + (8 − 10.4)2 + (11 − 10.4)2 + (15 − 10.4)2
=
5−1
r
103.2
=
4
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
11 / 23
Range & Standard Deviation of a Data Set (Example)
WEX 14-3-1: Given the following data set:
3, 15, 8, 11, 15
(a) Compute the range of the data set.
(b) Compute the mean of the data set.
(c) Compute the standard deviation of the data set.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 15 − 3 = 12
P
x
3 + 15 + 8 + 11 + 15
52
(b) x =
=
=
= 10.4
n
5
5
s
P
2
(x − x)
(c) s =
n−1
r
(3 − 10.4)2 + (15 − 10.4)2 + (8 − 10.4)2 + (11 − 10.4)2 + (15 − 10.4)2
=
5−1
r
103.2
=
≈ 5.0794
4
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
12 / 23
Range & Standard Deviation of a Freq. Dist. (Def’n)
Definition
(Range of a Frequency Distribution)
The range is the difference between the largest and smallest values.
Definition
(Standard Deviation of a Frequency Distribution)
The standard deviation measures ”how spread out” the data is:
v h
i
uP
2
u
(x − x) · f
t
P
s=
( f) − 1
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
13 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Compute the range of the frequency distribution.
(b) Compute the mean of the frequency distribution.
(c) Compute the standard deviation of the frequency distribution.
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
14 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Compute the range of the frequency distribution.
(b) Compute the mean of the frequency distribution.
(c) Compute the standard deviation of the frequency distribution.
(a) Range = (Largest Data Value) − (Smallest Data Value)
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
15 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Compute the range of the frequency distribution.
(b) Compute the mean of the frequency distribution.
(c) Compute the standard deviation of the frequency distribution.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 20 − 6 = 14
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
16 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Compute the range of the frequency distribution.
(b) Compute the mean of the frequency distribution.
(c) Compute the standard deviation of the frequency distribution.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 20 − 6 = 14
P
(x · f )
(b) x = P
f
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
17 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Compute the range of the frequency distribution.
(b) Compute the mean of the frequency distribution.
(c) Compute the standard deviation of the frequency distribution.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 20 − 6 = 14
P
(x · f )
(6)(23) + (9)(12) + (20)(31)
(b) x = P
=
f
23 + 12 + 31
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
18 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Compute the range of the frequency distribution.
(b) Compute the mean of the frequency distribution.
(c) Compute the standard deviation of the frequency distribution.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 20 − 6 = 14
P
(x · f )
(6)(23) + (9)(12) + (20)(31)
866
(b) x = P
=
=
≈ 13.1212
f
23 + 12 + 31
66
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
19 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Compute the range of the frequency distribution.
(b) Compute the mean of the frequency distribution.
(c) Compute the standard deviation of the frequency distribution.
(a) Range = (Largest Data Value) − (Smallest Data Value) = 20 − 6 = 14
P
(x · f )
(6)(23) + (9)(12) + (20)(31)
866
(b) x = P
=
=
≈ 13.1212
f
23 + 12 + 31
66
v h
i
uP
2
u
(x − x) · f
t
P
(c) s =
( f) − 1
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
20 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Range = (Largest Data Value) − (Smallest Data Value) = 20 − 6 = 14
P
(x · f )
(6)(23) + (9)(12) + (20)(31)
866
=
=
≈ 13.1212
(b) x = P
f
23 + 12 + 31
66
v h
i
uP
2
u
(x
−
x)
·
f
t
P
(c) s =
( f) − 1
r
23(6 − 13.1212)2 + 12(9 − 13.1212)2 + 31(20 − 13.1212)2
=
66 − 1
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
21 / 23
Range & Standard Dev. of a Freq. Dist. (Example)
WEX 14-3-2: Given the following frequency distribution:
DATA VALUE
(x)
6
9
20
FREQUENCY
(f )
23
12
31
(a) Range = (Largest Data Value) − (Smallest Data Value) = 20 − 6 = 14
P
(x · f )
(6)(23) + (9)(12) + (20)(31)
866
(b) x = P
=
=
≈ 13.1212
f
23 + 12 + 31
66
v h
i
uP
2
u
x)
·
f
(x
−
t
P
(c) s =
( f) − 1
r
23(6 − 13.1212)2 + 12(9 − 13.1212)2 + 31(20 − 13.1212)2
=
66 − 1
r
2837.03
=
= 6.607
65
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
22 / 23
Fin
Fin.
Josh Engwer (TTU)
Statistics: Range, Standard Deviation
03 August 2015
23 / 23