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Descriptive Statistical Tools
Outline of Discussion
Lesson Proper
◦ Descriptive Statistics
Data Organization
Data Analysis
Statistical Measures
Case Study
Outline of Discussion
Lesson Proper
◦ Descriptive Statistics
Data Organization
Data Analysis
Statistical Measures
Case Study
Descriptive Statistics
What is Descriptive Statistics?
◦ Also known as deductive statistics
◦ Deals with gathering, classification, and
presentation of data
◦ Summarizes values to describe group
characteristics of data
Descriptive Statistics
Data Presentation
◦ Presenting data in tabular or graphical form is
not enough to get all the relevant information
◦ Data must be organized and analysis must be
readily made
Data organization tools
Frequency distribution table, histogram, ogive
Data analysis tools
Stem-and-leaf diagram, boxplot, time-series plot, probability
plot, scatter plot
Descriptive Statistics
Data Presentation
◦ The common statistics included may also not
be adequate to describe data
◦ There are many more measures used
Measures of central tendency
Mean, trimmed mean, median, mode
Measures of dispersion
Standard deviation, variance, range, interquartile range, mean
absolute deviation, coefficient of dispersion
Measures on individual data points
Standard deviation unit, standard score
Descriptive Statistics
Data Presentation
◦ The common statistics included may also not
be adequate to describe data
◦ There are many more measures used
Measures of location
Percentile
Measures of skewness and kurtosis
Coefficient of skewness, coefficient of peakedness
Measure of linear relationship
Correlation coefficient, Pearson’s r coefficient
Outline of Discussion
Lesson Proper
◦ Descriptive Statistics
Data Organization
Data Analysis
Statistical Measures
Case Study
Descriptive Statistics
Data Organization
◦ Frequency Distribution Table
How to group data
Original data presention
Number of cells
Cell width
Cell boundaries
Long Exam 1 Scores
71
48
...
25
65
...
75
53
...
75
51
...
25
65
...
5
63
...
75
66
...
25
66
...
75
67
...
75
54
...
25
42
...
5
67
...
25
61
...
75
66
...
75
49
...
75
60
...
5
54
...
25
58
...
75
56
...
25
63
...
25
55
...
5
49
...
5
59
...
5
69
...
5
79
...
5
49
...
5
59
...
5
69
...
5
79
...
5
49
...
5
59
...
5
69
...
5
79
...
5
49
...
5
59
...
5
69
...
5
79
...
5
49
...
5
59
...
5
69
...
5
79
...
5
49
...
5
59
...
5
69
...
5
79
...
5
49
...
5
59
...
5
69
...
5
79
...
0943
0
...
1509
0
...
1887
0
...
0755
0
...
0943
0
...
3585
0
...
7547
0
...
9623
1
Descriptive Statistics
Data Organization
◦ Cumulative Frequency Distribution Table
Can answer the ff
...
5
49
...
5
59
...
5
69
...
5
79
...
5
49
...
5
59
...
5
69
...
5
79
...
0943
0
...
1509
0
...
1887
0
...
0755
0
...
0943
0
...
3585
0
...
7547
0
...
9623
1
Descriptive Statistics
Data Organization
◦ Ogive
Line graph of CFD
Cell Boundaries
[42 - 47)
[47 - 52)
[52 - 57)
[57 - 62)
[62 - 67)
[67 - 72)
[72 - 77)
[77 - 82]
For ≤ CFD, x-axis is upper cell boundaries
For ≥ CFD, x-axis is lower cell boundaries
Y-axis is cumulative frequency
Y-axis can also be relative frequency
Cumulative Frequency Distribution Table
fi
xi
rel fi
≤ CFD
≥ CFD
5
6
8
11
10
7
4
2
44
...
5
54
...
5
64
...
5
74
...
5
0
...
1132
0
...
2075
0
...
1321
0
...
0377
5
11
19
30
40
47
51
53
53
48
42
34
23
13
6
2
rel CFD
0
...
2075
0
...
566
0
...
8868
0
...
5
49
...
5
59
...
5
69
...
5
79
...
0943
0
...
1509
0
...
1887
0
...
0755
0
...
0943
0
...
3585
0
...
7547
0
...
9623
1
Ogive for relative CFD
1
...
8
0
...
4
0
...
5
49
...
5
59
...
5
69
...
5
79
...
0943
0
...
1509
0
...
1887
0
...
0755
0
...
0943
0
...
3585
0
...
7547
0
...
9623
1
Ogive for ≥ CFD
60
50
40
30
20
10
0
42
47
52
57
62
67
72
77
Descriptive Statistics
Data Organization
◦ Ogive
Line graph of CFD
Cell Boundaries
[42 - 47)
[47 - 52)
[52 - 57)
[57 - 62)
[62 - 67)
[67 - 72)
[72 - 77)
[77 - 82]
For ≤ CFD, x-axis is upper cell boundaries
For ≥ CFD, x-axis is lower cell boundaries
Y-axis is cumulative frequency
Y-axis can also be relative frequency
Cumulative Frequency Distribution Table
fi
xi
rel fi
≤ CFD
≥ CFD
5
6
8
11
10
7
4
2
44
...
5
54
...
5
64
...
5
74
...
5
0
...
1132
0
...
2075
0
...
1321
0
...
0377
5
11
19
30
40
47
51
53
53
48
42
34
23
13
6
2
rel CFD
0
...
2075
0
...
566
0
...
8868
0
...
g
...
g
...
5
Get quartile 3 (75th percentile)
V Quartile 1 = 68, W Quartile 1 = 68
...
5
V Quartile 3 = 68, W Quartile 3 = 68
...
g
...
)
Y-axis: value of variable
Descriptive Statistics
Data Analysis
◦ Time-series Plot
How to interpret a time-series plot
Example: Average weight of manufactured 100g potato chip
packs per day
Mean is around 100g
No pattern meaning random
Considering acceptance limits, a few less than 97g or more than 103g
are rejected
Time-series Plot
106
Average Weight
104
102
100
98
96
94
92
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29
Day
Descriptive Statistics
Data Analysis
◦ Time-series Plot
How to interpret a time-series plot
Example: Average weight of manufactured 100g potato chip
packs per day of another company
Mean is around 100g (at first)
Downward trend (something might be wrong with manufacturing)
Considering acceptance limits, many are being rejected
Time-series Plot
Average Weight
104
102
100
98
96
94
92
90
88
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29
Day
Descriptive Statistics
Data Analysis
◦ Time-series Plot
How to interpret a time-series plot
Example: Monthly sales of jackets of an apparel store
Mean is around 80 units
Cyclic pattern (there might be a reason to this cycle)
Monthly sales of jackets increase when nearing 12th and 24th month
(because people are buying more jackets during the Ber months)
Time-series Plot
120
100
Sales
80
60
40
20
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Month
Descriptive Statistics
Data Analysis
◦ Time-series Plot
How to interpret a time-series plot
Example: Price of ABS-CBN shares over 20 years
Descriptive Statistics
Data Analysis
◦ Time-series Plot
How to interpret a time-series plot
Example: Price of TEL (PLDT) shares over 5 years
Descriptive Statistics
Data Analysis
◦ Probability Plot
How to construct a probabi
Unit is unit2 of the variable
Descriptive Statistics
Statistical Measures
◦ Measures of Variability/Dispersion
Standard deviation
Most commonly used measure of variability/dispersion
Deviation of data from the mean
Always positive
Square root of variance
Unit is same as that of the variable
Descriptive Statistics
Statistical Measures
◦ Relative measure of Variability/Dispersion
Coefficient of dispersion
Used to compare different populations
The lesser the value, the more consistent the data
s is the sample standard deviation
x is the sample mean
Descriptive Statistics
Statistical Measures
◦ Relative measure of Variability/Dispersion
Example:
Determine who among two friends, A and B, have more
consistent sleeping hours
Quiz Question
Sleeping hours
Friend A
Friend B
xbar
9
6
...
5
Hint: Solve for each friend’s coefficient of variability
Descriptive Statistics
Statistical Measures
◦ Measures on Individual Data Points
Standard deviation unit
Distance of a point from the mean
The lesser the value, the closer the point is to the mean
s is the sample standard deviation
xi is the certain point in subject
xbar is the sample mean (or any point of reference)
Descriptive Statistics
Statistical Measures
◦ Measure of Symmetry
Coefficient of skewness
Determines the symmetry of a distribution
Third moment about the mean
xbar is the sample mean
n is the total number of data points
s is the sample standard deviation
a3 = 0; symmetric data set
a3 < 0; skewed to the left data set
a3 > 0; skewed to the right data set
Descriptive Statistics
Statistical Measures
◦ Measure of Symmetry
Coefficient of skewness
Descriptive Statistics
Statistical Measures
◦ Measure of Kurtosis
Coefficient of peakedness
Determines the height of a unimodal distribution
xbar is the sample mean
n is the total number of data points
s is the sample standard deviation
a4 = 3; data is mesokurtic (normal)
a4 > 3; data is leptokurtic (high peakedness)
a4 < 3; data is platykurtic (low peakedness)
Descriptive Statistics
Statistical Measures
◦ Measure of Kurtosis
Coefficient of peakedness
Descriptive Statistics
Statistical Measures
◦ Measure of Linear Relationship
Correlation coefficient
Determines the linearity between variables of a population
Pearson’s r coefficient
Determines the linearity between variables of a sample
Nonzero correlation coefficient means there is a linear relationship
Zero correlation coefficient means either they are independent of each
other or their relationship is nonlinear
Excel scatterplot uses r2 which is more accurate and reliable
Outline of Discussion
Lesson Proper
◦ Descriptive Statistics
Data Organization
Data Analysis
Statistical Measures
Case Study
Summary
Data Organization
◦
◦
◦
◦
◦
Frequency Distribution Table
Histogram
Cumulative Frequency Distribution Table
Ogive
Pareto Chart
Summary
Data Analysis
◦
◦
◦
◦
◦
Stem and Leaf Diagram
Boxplot
Time-series Plot
Probability Plot
Scatter Plot
Outline of Discussion
Lesson Proper
◦ Descriptive Statistics
Data Organization
Data Analysis
Statistical Measures
Case Study
CASE STUDY
For Each Data Set, Get the following:
◦
◦
◦
◦
◦
◦
◦
◦
Mean
Median
Mode
25th & 75th Percentile
Skewness
Kurtosis
Histogram
Ogives
Which Data sets are skewed?
Which Data sets are leptokurtic?
For each Data Set which measure of
Central Tendency is more appropriate to
be used when drawing conclusions?
Suppose that Data Set 2 is data used for
arm’s reach for Filipinos
...
95% of the Filipinos must be able
to reach this emergency button, how far
should the emergency button be?
A hazardous chemical shelf is to be
installed