Download 4.1.1.A Statistics

Statistics
Principles of Engineering
© 2012 Project Lead The Way, Inc.
Statistics
The collection, evaluation, and interpretation
of data
Statistics
Statistics
Descriptive Statistics
Inferential Statistics
Describe collected data
Generalize and
evaluate a population
based on sample
data
Data
Categorical or Qualitative Data
Values that possess names or labels
Color of M&M’s, breed of dog, etc.
Numerical or Quantitative Data
Values that represent a measurable quantity
Population, number of M&M’s, number
of defective parts, etc.
Data Collection
Sampling
• Random
• Systematic
• Stratified
• Cluster
• Convenience
Graphic Data Representation
Histogram
Frequency distribution graph
Frequency Polygons
Frequency distribution graph
Bar Chart
Categorical data graph
Pie Chart
Categorical data graph %
Measures of Central Tendency
Mean x
• Arithmetic average
• Sum of all data values divided by the
number of data values within the array
Sx
x=
n
• Most frequently used measure of central
tendency
• Strongly influenced by outliers—very
large or very small values
Measures of Central Tendency
Determine the mean value of
48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55
Sx
x=
n
(48+63+62+49+58+2+63+5+60+59+55)
x=
11
524
x=
11
x = 47.64
Measures of Central Tendency
Median
• Data value that divides a data array into
two equal groups
• Data values must be ordered from
lowest to highest
• Useful in situations with skewed data
and outliers (e.g., wealth
management)
Measures of Central Tendency
Determine the median value of
48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55
Organize the data array from lowest to
highest value.
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
Select the data value that splits the data set
evenly.
Median = 58
What if the data array had an even number of
values?
5, 48, 49, 55, 58, 59, 60, 62, 63, 63
Measures of Central Tendency
Mode
• Most frequently occurring response
within a data array
• Usually the highest point of curve
• May not be typical
• May not exist at all
• Modal, bimodal, and multimodal
Measures of Central Tendency
Determine the mode of
48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55
Mode = 63
Determine the mode of
48, 63, 62, 59, 58, 2, 63, 5, 60, 59, 55
Mode = 63 & 59 Bimodal
Determine the mode of
48, 63, 62, 59, 48, 2, 63, 5, 60, 59, 55
Mode = 63, 59, & 48
Multimodal
Data Variation
Measure of data scatter
Range
Difference between the lowest and highest
data value
Standard Deviation
Square root of the variance
Range
Calculate by subtracting the lowest value
from the highest value.
R=h-l
Calculate the range for the data array.
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
R=h-l
R = 63 - 2
R = 61
Standard Deviation – Sample vs. Population
Sample Standard Deviation
s=
( )
S x-x
(n-1)
Population Standard Deviation.
2
σ=
xi − μ
N
2
In practice, only the sample standard deviation can be measured and therefore is more
useful for applications.
Population Standard Deviation
A population standard deviation represents a parameter, not a statistic. The standard
deviation of a population gives researchers an amount of dispersion of data for an entire
population of survey respondents.
Sample Standard Deviation
A standard deviation of a sample estimates the standard deviation of a population based
on a random sample. The sample standard deviation, unlike the population standard
deviation, is a statistic that measures the dispersion of the data around the sample
mean.
Sample Standard Deviation
s for a sample, not population
1. Calculate the mean x
s=
( )
S x-x
(n-1)
2. Subtract the mean from each value and then
square it.
3. Sum all squared differences.
4. Divide the summation by the number of
values in the array minus 1.
5. Calculate the square root of the product.
2
Sample Standard Deviation
S (x-x )
Calculate the sample standard s=
(n-1)
deviation for the data array.
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
Sx
1. x =
n
2
2. (x - x )
524
=
11
=47.64
(2 - 47.64)2 = 2083.01
(59 - 47.64)2 = 129.05
(5 - 47.64)2 = 1818.17
(60 - 47.64)2 = 152.77
(48 - 47.64)2 =
0.13
(62 - 47.64)2 = 206.21
(49 - 47.64)2 =
1.85
(63 - 47.64)2 = 235.93
(55 - 47.64)2 =
54.17
(63 - 47.64)2 = 235.93
(58 - 47.64)2 = 107.33
2
Sample Standard Deviation
Calculate the standard
deviation for the data array.
2
s=
S (x-x )
(n-1)
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
2
4. S (x-x )
2083.01 + 1818.17 + 0.13 + 1.85 + 54.17 + 107.33
+ 129.05 + 152.77 + 206.21 + 235.93 + 235.93
= 5,024.55
2
5. S (x-x ) = 5,024.55 =502.46
(n-1)
10
2
6.
s=
S (x-x )
(n-1)
= 502.46
s = 22.42
Population Standard Deviation
Calculate the population standard
deviation for the data array
σ=
xi − μ
N
2
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
xi
524
1. Calculate the mean

= 47.64
μ=
11
N
2. Subtract the mean from each data value and square each
2
difference
xi − μ
(2 - 47.63)2 = 2082.6777
(5 - 47.63)2 = 1817.8595
(48 - 47.63)2 =
0.1322
(49 - 47.63)2 =
1.8595
(55 - 47.63)2 = 54.2231
(58 - 47.63)2 = 107.4050
(59 - 47.63)2 =
(60 - 47.63)2 =
(62 - 47.63)2 =
(63 - 47.63)2 =
(63 - 47.63)2 =
129.1322
152.8595
206.3140
236.0413
236.0413
Population Standard Deviation
Variation
3. Sum all squared differences
2
2082.6777 + 1817.8595 + 0.1322 + 1.8595 + 54.2231 +
xi − μ =
107.4050 + 129.1322 + 152.8595 + 206.3140
+ 236.0413 + 236.0413
= 5,024.5455
Note that this is the sum of the
unrounded squared differences.
4. Divide the summation by the number of data values
2
xi − μ
5024.5455
=
= 456.7769
N
11
5. Calculate the square root of the result
xi − μ
N
2
= 456.7769 = 21.4
Graphing Frequency Distribution
Numerical assignment of each outcome of a
chance experiment
A coin is tossed three times. Assign the
variable X to represent the frequency of
heads occurring in each toss.
Toss Outcome
HHH
x Value
3
2
x =1 when?
HTH
THH
2
HTT,THT,TTH
HTT
THT
1
1
TTH
1
TTT
0
HHT
2
Graphing Frequency Distribution
The calculated likelihood that an outcome
variable will occur within an experiment
Toss Outcome
X value
HHH
3
2
HHT
HTH
THH
2
HTT
THT
1
1
TTH
1
TTT
0
2
x
0
fx
Px =
fa
1
2
3
Px
P0 =
1
8
3
8
3
P2 =
8
P1=
P3 =
1
8
Graphing Frequency Distribution
Histogram
x
0
1
2
3
Px
P0 =
1
8
3
8
3
P2 =
8
P1=
P3 =
1
8
x
Histogram
Available airplane passenger seats one week
before departure
percent of the time
What information does
the histogram provide
the airline carriers?
What information
does the histogram
provide prospective
customers?
open seats