Transcript Results and analysis 1

RESEARCH METHODOLOGY
RESULT AND ANALYSIS
(part 1)
Introduction
DATA ANALYSIS is a process of inspecting,
cleaning, transforming, and modeling data
with the goal of highlighting useful
information, suggesting conclusions, and
supporting decision making.
Data analysis has multiple facets and
approaches, encompassing diverse
techniques under a variety of names, in
different business, science, and social
science domains.
Type of data
 Quantitative data. data is a number
 Often this is a continuous decimal number to
a specified number of significant digits
 Sometimes it is a whole counting number
 Categorical data. data one of several categories
 Qualitative data. data is a pass/fail or the
presence or lack of a characteristic
Quantitative data
 Quantitative data is data measured or identified on a
numerical scale. Numerical data can be analyzed using statistical
methods, and results can be displayed using tables, charts,
histograms and graphs.
Examples of quantitative data would be:
 Counts
 'there are 643 dots on the ceiling'
 'there are 25 pieces of bubble gum'
 'there are 8 planets in the solar system'
 Measurements
 'the length of this table is 1.892m'
 'the temperature at 12:00 p.m. was 18.9° Celsius'
 'the average flow yesterday in this river was 25 mph (miles per hour)'
Categorical data
 Categorical data is that part of an observed dataset that
consists of categorical variables, or for data that has been
converted into that form, for example as grouped data.
Example
The idea of grouped data can be illustrated by considering
the following raw dataset:
Table 1: Time taken (in seconds) by a group of students to
answer a simple math question
20
26
16
14
25
8
21
15
24
19
17
21
33
31
11
18
13
11
34
17
The above data can be organized into a frequency distribution (or a
grouped data) in several ways. One method is to use intervals as a basis.
The smallest value in the above data is 8 and the largest is 34.
Table 2: Frequency distribution of the time taken (in seconds) by the group of
students to answer a simple math question
Time taken (in seconds)
5 and above, below 10
10 and above, below 15
15 and above, below 20
20 and above, below 25
25 and above, below 30
30 and above, below 35
Frequency
1
4
6
4
2
3
Another method of grouping the data is to use some
qualitative characteristics instead of numerical intervals.
Table 3: Frequency distribution of the three types of students
Frequency
Smart
5
Normal
10
Below normal
5
Qualitative data
 Qualitative data describe items in terms of some quality or
categorization that in some cases may be 'informal‘
 In regression analysis, dummy variables are a type of qualitative
data.
 For example, if various features are observed about each of various
human subjects, one such feature might be gender, in which case a
dummy variable can be constructed that equals 0 if the subject is
male and equals 1 if the subject is female. Then this dummy
variable can be used as an independent variable (explanatory
variable) in an ordinary least squares regression. Dummy variables
can also be used as dependent variables, in which case the probit
or logistic regression technique would typically be used.
Quality of data
 The quality of the data should be checked as early as
possible.
 Data quality can be assessed in several ways, using
different types of analyses: frequency counts,
descriptive statistics (mean, standard deviation,
median), normality (skewness, kurtosis, frequency
histograms, normal probability plots), associations
(correlations, scatter plots).
Data analysis tools
Commonly used approaches or tools
 Statistics
 Models
 Standards
Statistic
 Statistics is the study of the collection, organization,
analysis, and interpretation of data
Modelling
Data modeling is a method used to define and
analyze data requirements needed to support the
business processes of an organization.
Standard
 American Measurement Standard (AMS)
 Deutsches Institut für Normung (DIN; in English,
the German Institute for Standardization)
 International Standard Organization (ISO)
 Australia Standards
 Institute for Reference Materials and
Measurements (EU)
Statistical Analysis
Two main areas of statistics
 Descriptive statistics. summarize the population data by describing what
was observed in the sample numerically or graphically. Numerical
descriptors include mean and standard deviation for continuous data types
(like heights or weights), while frequency and percentage are more useful in
terms of describing categorical data (like race). Involved : data collection,
organization, summation
 Inferential statistics. uses patterns in the sample data to draw inferences
about the population represented, accounting for randomness. These
inferences may take the form of: answering yes/no questions about the data
(hypothesis testing), estimating numerical characteristics of the data
(estimation), describing associations within the data (correlation) and
modeling relationships within the data (for example, using regression
analysis). generalizing from samples to populations. Involved: performing
hypothesis testing, determining relationships among variables, and making
predictions
DATA DESCRIPTION
Three aspects:
1. Measures of Central Tendency
Mean
Median
Mode
Midrange
Definition
sum of values divided by
total number of value
Middle point in the data
set
Most frequent data value
(Lowest value plus highest
value)/2
Symbol
, x
MD
None
MR
2. Measures of Variation.
Sometime the mean is not good enough to describe a data
set as in the following example.
Example: A testing lab wishes to test two experimental
brands of outdoor paint to see how long each would last
before fading. Different chemical agents are added in
each group and only six cans are involved. These two
groups constitute two small populations. The results (in
months) follow.
Brand A
10
60
50
30
40
20
Mean = 35
Brand B
35
45
30
35
40
25
Mean 35
Note that Brand A and B gave similar means = 35.
Thus one might conclude that both brand of paint last
equally well. But a different conclusion might be
withdrawn when the data set are examined graphically.
The range for Brand A: 60-10 = 50 month
for Brand B: 45-25 = 20 month
Measures indicating the degree of spread/variation
Range
Definition
Symbols
distance between highest and lowest value
R
Variance
average of the squares of the distance
each value id from the mean
Standard
Deviation
Square root of the variance
2, s2
, s
3. Measure of Position
Definition
Symbol
Standard score Number of standard
z
or z score
deviation a data value is
above or below the mean
Percentile
Position in hundredths a
Pn
data value is in the
distribution
Decile
Position in tenths a data
Dn
values is in the distribution
Quartile
Position in fourths a data
Qn
value is in the distribution
Mode
 The mode is the most repeated value in a distribution.
 It is represented by Mo.
 It is possible to find the mode for categorical and quantitative
variables.
Median
 The median is the score of the scale that separates the upper half
of the distribution from the lower, that is to say, it divides the
series of data into two equal parts.
 The median is denoted by Me.
 The median can only be found for quantitative variables.
Calculation of the Median for Grouped Data
Mean
In statistics, mean has two related meanings:
 the arithmetic mean (and is distinguished from the geometric
mean or harmonic mean).
 the expected value of a random variable, which is also called the
population mean.
The arithmetic mean is the "standard" average, often
simply called the "mean".
 For example, the arithmetic mean of six values: 34, 27, 45,
55, 22, 34 is
Geometric mean (GM)
The geometric mean is an average that is useful for sets of
positive numbers that are interpreted according to their
product and not their sum (as is the case with the
arithmetic mean) e.g. rates of growth.
For example, the geometric mean of six values: 34,
27, 45, 55, 22, 34 is:
Harmonic mean (HM)
The harmonic mean is an average which is useful for
sets of numbers which are defined in relation to some
unit, for example speed (distance per unit of time).
For example, the harmonic mean of the six
values: 34, 27, 45, 55, 22, and 34 is
Population mean
The mean of a population has an expected value of
μ, known as the population mean. The sample mean
makes a good estimator of the population mean, as
its expected value is the same as the population
mean. The sample mean of a population is a random
variable, not a constant, and consequently it will
have its own distribution. For a random sample of n
observations from a normally distributed population,
the sample mean distribution is
Standard deviation
 It shows how much variation or "dispersion" there is from the
average (mean, or expected value).
Consider a population consisting of the following eight values:
These eight data points have the mean (average) of 5:
To calculate the population standard deviation, first compute the
difference of each data point from the mean, and square the result of
each:
Next compute the average of these values, and take the square root:
Example:
Calculate the standard deviation for the following sample data using all
methods: 2, 4, 8, 6, 10, and 12.
Solution:
Percentile
 percentile (or centile) is the value of a variable below
which a certain percent of observations fall.
One definition of percentile, often given in texts, is that the
P-th percentile ( ) of N ordered values (arranged from least
to greatest) is obtained by first calculating the (ordinal) rank
rounding the result to the nearest integer, and then taking
the value that corresponds to that rank.
For example, by this definition, given the numbers
15, 20, 35, 40, 50
the rank of the 30th percentile would be
.
Thus the 30th percentile is 20, the second number in the
sorted list.
The 40th percentile would have rank
,
Percentile
Quartiles
 quartiles of a set of values are the three points that divide
the data set into four equal groups, each representing a
fourth of the population being sampled.
1. first quartile (designated Q1) = lower quartile = cuts
off lowest 25% of data = 25th percentile
2. second quartile (designated Q2) = median = cuts data
set in half = 50th percentile
3. third quartile (designated Q3) = upper quartile =
cuts off highest 25% of data, or lowest 75% = 75th
percentile
Exploratory data analysis
 exploratory data analysis (EDA) is an approach to
analysing data set to summarize their main
characteristics in easy-to-understand form, often with
visual graphs, without using a statistical model or having
formulated a hypothesis.
 To discover various aspects of data. In EDA data are are
organised to facilitate further analysis
Common methods used
1. Stem and leaf plot
2. Box Plots
Stem-and-leaf display
 A stemplot (or stem-and-leaf display), in statistic, is a device
for presenting quantitative data in a grapical format, to assist in
visualizing the shape of a distribution.
44 46 47 49 63 64 66 68 68 72 72 75 76 81 84 88 106
Box plot
 box plot or boxplot (also
known as a box-and-whisker
diagram or plot) is a
convenient way of graphically
depicting groups of numerical
data through their five-number
summaries: the smallest
observation (sample minimum),
lower quartile (Q1), median
(Q2), upper quartile (Q3), and
largest observation (sample
maximum).
Information Obtained from a
Box Plot
a. If the median is near the center of the box, the
distribution is approximately symmetric
b. If the median falls to the left of the center of the
box, the distribution is positively skewed
c. If the median falls to the right of the center, the
distribution is negatively skewed
d. If the lines are about the same length, the
distribution is approximately symmetric
e. If the right line is larger than the left line, the
distribution is positively skewed
f. If the left line is larger than the right line, the
distribution is negatively skewed
GRAPHICAL PRESENTATION
The most commonly used graphs
 bar chart or bar graph is a chart with rectangular bars
with lengths proportional to the values that they represent.
The bars can be plotted vertically or horizontally.
Histogram
 histogram is a graphical representation showing a visual
impression of the distribution of data.
Histogram
 A histogram is a graphic representation of a variable in the
shape of bars (rectangles).
 They are used for continuous or discrete variables with a
large quantity of data that is grouped into classes.
 The base width of the bars (rectangles) are proportional
to the class widths and the height is the absolute
frequency of each interval.
 The surface area of every bar is proportional to the frequency
of the represented values.
Run chart
 run-sequence plot is a graph that displays observed data in
a time sequence.
Pie chart
 A pie chart can be used to represent all types of variables,
but is more commonly used for categorical variables.
 The data is represented in a circle and the angle of each
circular sector is proportional to the corresponding
absolute frequency.
 The pie chart can be constructed with the help of a
protractor.
Radar chart
 radar chart is a graphical method of displaying multivariate
data in the form of a two-dimensional chart of three or more
quantitative variables represented on axes starting from the
same point.
Scatter plot
 scatter plot or scattergraph is a type of mathematical
diagram using Cartesian coordinates to display values for
two variables for a set of data.
Problem
 Given the data set
4 , 10 , 7 , 7 , 6 , 9 , 3 , 8 , 9
Find
a) the mode,
b) the median,
c) the mean (Arithmetic, Geometric and Harmonic)
d) the sample standard deviation.
e) If we replace the data value 6 in the data set above by 24, will the
standard deviation increase, decrease or stay the same?
Solution
 The given data set has 2 modes: 7 and 9
 order data : 3 , 4 , 6 , 7 , 7 , 8 , 9 , 9 , 10 : median = 7
 (mean) : m = (3+4+6+7+7+8+9+9+10) / 9 = 7
Problem
 Given the data set
62 , 65 , 68 , 70 , 72 , 74 , 76 , 78 , 80 , 82 , 96 , 101,
find
a) the median,
b) the first quartile,
c) the third quartile,
c) the interquartile range (IQR).
Solution
 median = 75
 first quartile = 69
 third quartile = 81
 interquartile range = 81 - 69 = 12