Math 105: Engineering Data Analysis
CUMULATIVE FREQUENCY DISTRIBUTIONS
 The cumulative frequency is the total frequency of all values either
“less than” or “more than” any class boundary.
CUMULATIVE FREQUENCY DISTRIBUTION OF GASOLINE
CONSUMPTION:
Gasoline Consumption
mi/gal
“less than” Cumulative
Frequency
Gasoline Consumption
mi/gal
“more than”
Cumulative Frequency
Less than 22.45
0
More than 22.45
40
Less than 22.95
2
More than 22.95
38
Less than 23.45
5
More than 23.45
35
Less than 23.95
12
More than 23.95
28
Less than 24.45
20
More than 24.45
20
Less than 24.95
34
More than 24.95
6
Less than 25.45
39
More than 25.45
1
Less than 25.95
40
More than 25.95
0
Math 105: Engineering Data Analysis
CUMULATIVE FREQUENCY DISTRIBUTION OF COMPRESSIVE
STRENGTH OF FIBER ROOF TILES:
Compressive Strength
Kg/cm2
“less than” Cumulative
Frequency
Compressive Strength
Kg/cm2
“more than” Cumulative
Frequency
Less than 79.5
0
More than 79.5
50
Less than 89.5
2
More than 89.5
48
Less than 99.5
5
More than 99.5
45
Less than 109.5
14
More than 109.5
36
Less than 119.5
27
More than 119.5
23
Less than 129.5
40
More than 129.5
10
Less than 139.5
47
More than 139.5
3
Less than 149.5
50
More than 149.5
0
Math 105: Engineering Data Analysis
HISTOGRAM,
FREQUENCY POLYGON
AND OGIVE
Math 105: Engineering Data Analysis
 A histogram, or frequency histogram, consists of a set of rectangles
having a.) bases equal to the class interval sizes with centers at the
class marks and b.) heights equal to the corresponding class
frequencies. The areas of the rectangles, therefore, are proportional
to the class frequencies.
 A frequency polygon is a line graph of class frequency plotted
against the class mark. It can also be obtained by connecting the
midpoints of the tops of the rectangle in the histogram.
 A line graph showing the cumulative frequency plotted against any
class boundary is called a cumulative frequency polygon or
ogive.
Math 105: Engineering Data Analysis
Math 105: Engineering Data Analysis
Math 105: Engineering Data Analysis
STEM AND LEAF DISPLAY
Stem and leaf display is a combined tabular and graphic display. The
following are suggested steps in constructing a stem and leaf display.
1. Select one or more leading digits for the stem values. The remaining
digits become the leaves.
2. List all the possible stem values in a vertical column.
3. Record the leaf for every observation beside the corresponding stem
value.
4. Indicate the units for stems and leaves someplace in the display.
A display having between 5 and 20 stems is recommended..
Math 105: Engineering Data Analysis
Example:
Construct the stem and leaf display of the data in your assignment.
136
92
115
118
121
137
132
120
104
125
119
115
101
129
87
108
110
133
135
126
127
103
110
126
118
82
104
134
120
95
146
126
119
119
105
132
126
118
100
113
106
125
117
102
146
129
124
113
95
148
Math 105: Engineering Data Analysis
NUMERICAL SUMMARY
OF MEASURES
Math 105: Engineering Data Analysis
PARAMETER AND STATISTICS
 Numerical summary measure focuses for qualitative data to
calculate a set of numbers that will characterize the data set and
convey some of its salient features. The important characteristics of
a set of numbers are its location, particularly the center, and its
variability.
 Numerical summary measures can be calculated from either a
sample or a population. Any quantitative measure that describes a
characteristics of a population is a parameter, it is a statistic when
it describes a characteristics of a sample.
Math 105: Engineering Data Analysis
Summation notations and rules with summary measures:
Math 105: Engineering Data Analysis
Examples:
1. The following data represents the time in seconds for 9 samples to
react to a stimulant: 2.5, 3.6, 3.1, 4.3, 2.9, 2.3, 2.6, 4.1 and 3.4.
Calculate the mean.
2. Find the mean weight of alum from the example in Module 1.
Weight (kl.)
No. of Bags
(frequency,f)
1.56
6
1.58
4
1.59
5
1.60
6
1.62
9
𝒏 = 𝟑𝟎
Math 105: Engineering Data Analysis
MEASURES OF CENTRAL
TENDENCY:
MEAN, MEDIAN, MODE
Math 105: Engineering Data Analysis
MEAN
 The arithmetic mean or mean is the over all average.
 If the data represent the entire population, the mean of the
values is referred to as the population mean, μ. If the data
constitute a sample drawn from a population, the mean referred to
as the sample mean, .
FORMULA:
Math 105: Engineering Data Analysis
MEAN OF GROUPED DATA
1. LONG METHOD
3. CODING METHOD
,
2. SHORT METHOD
,
A = any guessed or assumed class mark
∑
Math 105: Engineering Data Analysis
Example:
Find the mean of gasoline consumption from the example in Module 1.
Gasoline Consumption in miles/ gallon
No. of cars
(f requency,f)
22.5 - 22.9
2
23.0 – 23.4
3
23.5 – 23.9
7
24.0 – 24.4
8
24.5 – 24.9
14
25.0 – 25.4
5
25.5 – 25.9
1
sum
40
Math 105: Engineering Data Analysis
MEDIAN, x̃:
 The median of a set of numbers in an array is either the middle value
or the arithmetic mean of the two middle values.
If n is odd:
If n is even:
Math 105: Engineering Data Analysis
Examples:
1. Find the median of {1, 3, 3, 5, 6, 8, 9, 9, 10}
2. For the set of numbers {4,4,7,9,11,12,15,18}, find the median
3. Find the median of the data:
{2.5, 3.6, 3.1, 4.3, 2.9, 2.3, 2.6, 4.1, 3.4}
Math 105: Engineering Data Analysis
MEDIAN OF GROUPED DATA
Formula:
where :
Lm = lower class boundary of the median class
(Σf)L = sum of frequencies of all classes lower than the median class
fm = frequency of the median class
C = size of the median class
Math 105: Engineering Data Analysis
Example:
Determine the median of the data:
Math 105: Engineering Data Analysis
MEDIAN OF GROUPED DATA
Formula:
where :
Lm = lower class boundary of the median class
(Σf)L = sum of frequencies of all classes lower than the median class
fm = frequency of the median class
C = size of the median class
Math 105: Engineering Data Analysis
MODE,
 The value which occurs with greatest frequency in a dataset. Along
with mean and median, mode is a statistical measure of central
tendency in a dataset.
 Unlike the other measures of central tendency that are unique to a
particular dataset, there may be several modes in a dataset.
Steps to determine Mode:
1. Collect and organize the data from a dataset.
2. Determine all the distinct values in a dataset.
3. Count the frequency of occurrence for each distinct value.
4. The most frequent value(s) is the mode.
Math 105: Engineering Data Analysis
Examples:
1. For the set of numbers given, find the mode:
{3, 3, 5, 7, 9, 10, 11, 10, 11, 12, 9, 18, 9}
2. For the given dataset, find the mode:
{6, 7, 9, 10, 12}
3. For the given set of values, find the mode:
Math 105: Engineering Data Analysis
MODE OF GROUPED DATA
Formula:
where :
Lmo = lower class boundary of the modal class
d1 = excess of modal frequency of the next lower class
d2 = excess of modal frequency of the next higher class
c = size of the modal class interval
Math 105: Engineering Data Analysis
Example:
Find the mode of the data:
Math 105: Engineering Data Analysis
FREQUENCY CURVES AND SMOOTHED OGIVES
 In the population, so many observations are available so that it is
theoretically possible (for continuous data) to choose class intervals
from a very small. Thus, the frequency polygon for large population
have so many small broken line segments that they approximate
curves, which we call the frequency curves.
 By smoothing the frequency polygon of the sample, theoretical
curves of the population can be approximated.
In similar manner, smoothed
ogives are obtained by
smoothing the cumulative
frequency polygon.
Math 105: Engineering Data Analysis
TYPES OF FREQUENCY CURVES
1. Symmetrical or bell-shaped – when observations from the
central maximum have the same frequency.
2. Moderately asymmetrical or skewed – when the tail of the
curve to one side of the central maximum is longer than the other.
a. Skewed to the right – or to have positive skewness if the longer
tail occurs to the right of the central maximum.
b. Skewed to the left – or to have negative skewness when the
longer tail occurs to the left.
3. J-shaped or reverse J-shaped – maxima occurs at one end.
4. U-shaped – has maxima at both ends
5. Bimodal – has two maxima
6. Multimodal – has more than two maxima
Math 105: Engineering Data Analysis
THREE MEASURES OF
VARIABILITY
• RANGE
• VARIANCE
• STANDARD DEVIATION
Math 105: Engineering Data Analysis
RANGE
 defines the spread of data by calculating the difference between the
highest and lowest values in a dataset
 is the simplest measure of variability. It is the least satisfactory
because it provides no information at all about the data between
the highest and the lowest values.
Math 105: Engineering Data Analysis
VARIANCE
 To overcome the disadvantage of the range, the variability of the
data will depend upon the extent to which individual observations
are spread about the mean, since, it is by for the most important
measure of central tendency. The deviation of individual
observations are taken as deviation from the mean
.
 Variance is a measure that considers position of each observation
relative to the mean. It is defined as the squares of all the
deviations.
Math 105: Engineering Data Analysis
POPULATION VARIANCE
VERSUS
SAMPLEVARIANCE
Math 105: Engineering Data Analysis
Math 105: Engineering Data Analysis
Math 105: Engineering Data Analysis
Math 105: Engineering Data Analysis
Math 105: Engineering Data Analysis
Math 105: Engineering Data Analysis
Math 105: Engineering Data Analysis
Example:
The following reading were obtained for the tensile strength (in kg/cm)
of six specimens of an alloy:
2.58 2.65 2.40 2.46 2.44 2.41
Find the mean tensile strength and the standard deviation of the tensile
strengths.
Math 105: Engineering Data Analysis
Example:
Determine the standard deviation of gasoline consumption.