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Introduction to Summary Statistics
Statistics• The collection, evaluation, and interpretation of
data
• Statistical analysis of measurements can help verify the quality of a design or process
Summary Statistics
Central Tendency• “Center” of a distribution
– Mean, median, mode
Variation• Spread of values around the center
– Range, standard deviation, interquartile range
Distribution• Summary of the frequency of values
– Frequency tables, histograms, normal distribution
• The meanmean is the sum of the values of a set of data divided by the number of values in that data set.
Mean Central Tendency
Mean Central Tendency
• Data Set
3 7 12 17 21 21 23 27 32 36 44
• Sum of the values = 243
• Number of values = 11
Mean = 24311
= 22.09=
Mean Central Tendency
Mode Central Tendency
• Measure of central tendency
• The most frequently occurring value in a set of data is the mode
• Symbol is M
27 17 12 7 21 44 23 3 36 32 21
Data Set:
• The most frequently occurring value in a set of data is the mode
3 7 12 17 21 21 23 27 32 36 44
Data Set:
Mode = M = 21
Mode Central Tendency
• The most frequently occurring value in a set of data is the mode.
• Bimodal Data Set: Two numbers of equal frequency stand out
• Multimodal Data Set: If more than two numbers of equal frequency stand out
Mode Central Tendency
Determine the mode of
48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55Mode = 63
Determine the mode of
48, 63, 62, 59, 58, 2, 63, 5, 60, 59, 55Mode = 63 & 59 Bimodal
Determine the mode of
48, 63, 62, 59, 48, 2, 63, 5, 60, 59, 55Mode = 63, 59, & 48 Multimodal
Mode Central Tendency
• Measure of central tendency
• The median is the value that occurs in the middle of a set of data that has been arranged in numerical order
• Symbol is x, pronounced “x-tilde”~
Median Central Tendency
• The median is the value that occurs in the middle of a set of data that has been arranged in numerical order.
Data Set:27 17 12 7 21 44 23 3 36 32 21
Median Central Tendency
• A data set that contains an odd number of values always has a Median.
3 7 12 17 21 21 23 27 32 36 44Data Set:
Median Central Tendency
• For a data set that contains an even number of values, the two middle values are averaged with the result being the Median.
3 7 12 17 21 21 23 27 31 32 36 44Data Set:
Median Central Tendency
• Measure of data variation.
• The range is the difference between the largest and smallest values that occur in a set of data.
• Symbol is R
Range = R = 44 – 3 = 41
3 7 12 17 21 21 23 27 32 36 44Data Set:
Range Variation
• Measure of data variation.
• The standard deviation is a measure of the spread of data values.– A larger standard deviation indicates a wider
spread in data values
Standard Deviation Variation
Standard Deviation Variation
σ = standard deviation
xi = individual data value ( x1, x2, x3, …)
μ = mean
N = size of population
Standard Deviation Variation
Procedure:
1.Calculate the mean, μ.
2.Subtract the mean from each value and then square each difference.
3.Sum all squared differences.
4.Divide the summation by the size of the population (number of data values), N.
5.Calculate the square root of the result.
Standard Deviation
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
Calculate the standard deviation for the data array
524
111. Calculate the mean. 47.64
2. Subtract the mean from each data value and square each difference.
(2 - 47.64)2 = 2083.01 (5 - 47.64)2 = 1818.17(48 - 47.64)2 = 0.13(49 - 47.64)2 = 1.85(55 - 47.64)2 = 54.17(58 - 47.64)2 = 107.33
(59 - 47.64)2 = 129.05(60 - 47.64)2 = 152.77(62 - 47.64)2 = 206.21(63 - 47.64)2 = 235.93(63 - 47.64)2 = 235.93
Standard Deviation Variation
3. Sum all squared differences.
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
4. Divide the summation by the number of data values.
5. Calculate the square root of the result.
A Note about Standard Deviation
• Two distinct calculations– Population Standard Deviation
• The measure of the spread of data within a population.
• Used when you have a data value for every member of the entire population of interest.
– Sample Standard Deviation• An estimate of the spread of data within a larger
population.• Used when you do not have a data value for every
member of the entire population of interest.• Uses a subset (sample) of the data to generalize
the results to the larger population.
Population Standard Deviation
SampleStandard Deviation
A Note about Standard Deviation
σ = population standard deviationxi = individual data value ( x1, x2, x3, …)
μ = population mean
N = size of population
Sample Standard Deviation Variation
Sample Mean Central Tendency
Essen
tially
the
sam
e ca
lculat
ion a
s
popu
lation
mea
n
2. Subtract the sample mean from each data value and square the difference.
Sample Standard Deviation
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
Estimate the standard deviation for a population for which the following data is a sample.
524
11 47.641. Calculate the sample mean.
(2 - 47.64)2 = 2083.01 (5 - 47.64)2 = 1818.17(48 - 47.64)2 = 0.13(49 - 47.64)2 = 1.85(55 - 47.64)2 = 54.17(58 - 47.64)2 = 107.33
(59 - 47.64)2 = 129.05(60 - 47.64)2 = 152.77(62 - 47.64)2 = 206.21(63 - 47.64)2 = 235.93(63 - 47.64)2 = 235.93
Sample Standard Deviation Variation
= 5,024.55
3. Sum all squared differences.
2083.01 + 1818.17 + 0.13 + 1.85 + 54.17 + 107.33 + 129.05 + 152.77 + 206.21 + 235.93 + 235.93
4. Divide the summation by the number of sample data values minus one.
5. Calculate the square root of the result.
Population Standard Deviation
SampleStandard Deviation
A Note about Standard Deviation
σ = population standard deviationxi = individual data value ( x1, x2, x3, …)
μ = population mean
N = size of population
As n → N, s → σ
Population Standard Deviation
SampleStandard Deviation
A Note about Standard Deviation
σ = population standard deviationxi = individual data value ( x1, x2, x3, …)
μ = population mean
N = size of population
Given the ACT score of every student in your
class, use the population standard deviation formula to find the standard deviation of
ACT scoresin the class.
Population Standard Deviation
SampleStandard Deviation
A Note about Standard Deviation
σ = population standard deviationxi = individual data value ( x1, x2, x3, …)
μ = population mean
N = size of population
Given the ACT scores of every student in your
class, use the sample standard deviation formula to estimate the standard
deviation of the ACT scores of all students at
your school.
• A histogram is a common data distribution chart that is used to show the frequency with which specific values, or values within ranges, occur in a set of data.
• An engineer might use a histogram to show the variation of a dimension that exists among a group of parts that are intended to be identical.
Histogram Distribution
• Large sets of data are often divided into limited number of groups. These groups are called class intervals.
-5 to 5
Class Intervals6 to 16-6 to -16
Histogram Distribution
• The number of data elements in each class interval is shown by the frequency, which occurs along the Y-axis of the graph
Fre
qu
ency
1
3
5
7
-5 to 5 6 to 16-16 to -6
Histogram Distribution
3
ExampleF
req
uen
cy
1
2
4
6 to 10 11 to 151 to 5
1, 7, 15, 4, 8, 8, 5, 12, 10
12,15 1, 4, 5, 7, 8, 8, 10,
Histogram Distribution
• The height of each bar in the chart indicates the number of data elements, or frequency of occurrence, within each range
Histogram Distribution
3
Fre
qu
ency
1
2
4
6 to 10 11 to 151 to 5
12,15 1, 4, 5, 7, 8, 8, 10,
Class Intervals
MINIMUM = 0.745 in.
MAXIMUM = 0.760 in.
Histogram Distribution
0 1 2 3 4 5 6-1-2-3-4-5-6
0
3
-1
3
2
-1
-1
1
2
-3
0
1
0
1
-2
1
2
-4
-1
1
0
-2
0
0
Dot Plot Distribution
0 1 2 3 4 5 6-1-2-3-4-5-6
0
3
-1
3
2
-1
-1
1
2
-3
0
1
0
1
-2
1
2
-4
-1
1
0
-2
0
0
Fre
qu
ency
1
3
5
Dot Plot Distribution
“Is the data distribution normal?”
•Translation: Is the histogram/dot plot bell-shaped?
– Does the greatest frequency of the data values occur at about the mean value?
– Does the curve decrease on both sides away from the mean?
– Is the curve symmetric about the mean?
Normal Distribution Distribution
Fre
qu
ency
Data Elements
0 1 2 3 4 5 6-1-2-3-4-5-6
Bell shaped curve
Normal Distribution Distribution
Fre
qu
ency
Data Elements
0 1 2 3 4 5 6-1-2-3-4-5-6
Mean Value
Normal Distribution Distribution
Does the greatest frequency of the data values occur at about the mean value?
Fre
qu
ency
Data Elements
0 1 2 3 4 5 6-1-2-3-4-5-6
Mean Value
Normal Distribution Distribution
Does the curve decrease on both sides away from the mean?
Fre
qu
ency
Data Elements
0 1 2 3 4 5 6-1-2-3-4-5-6
Mean Value
Normal Distribution Distribution
Is the curve symmetric about the mean?
What if things are not equal?
Histogram Interpretation: Skewed (Non-Normal) Right
• 68% of the observations fall within 1 standard deviation of the mean.
• 95% of the observations fall within 2 standard deviations of the mean.
• 99.7% of the observations fall within 3 standard deviations of the mean.
Normal Distribution Distribution
If the data are normally distributed:
Normal Distribution ExampleData from a sample of a larger population
Data Elements
Normal Distribution Distribution
s +1.77
s -1.77
0.08
+ 1
.77
= 1
.88
0.08
+ -
1.7
7=
-1.
69
68 %
Data Elements
Normal Distribution Distribution
2σ - 3.54
2σ + 3.54
0.08
+ 3
.54
= 3
.62
0.08
+ -
3.54
=
- 3
.46
95 %