1. Slide 1
Lecture by
Dr Zahid Khan
King Faisal University,KSA.
1
Descriptive Statistics
2. Slide 2
4/20/2014
2
Types of Statistical Analyses
Five Types of Statistical Analysis:
1. Descriptive analysis: used to describe the data set
2. Inferential analysis: used to generate conclusions
about the population’s characteristics based on the
sample data
3. Slide 3
Descriptive Statistics
Tools for summarising, organising,
simplifying data.
1. Tables & Graphs
2. Measures of Central Tendency
3. Measures of Variability
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3
4. Slide 4
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4
Understanding Data Via Descriptive Analysis
Two sets of descriptive measures:
–Measures of central tendency: used to report a
single piece of information that describes the most
typical response to a question
–Measures of variability: used to reveal the typical
difference between the values in a set of values
5. Slide 5
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5
Understanding Data Via Descriptive Analysis
Measures of Central Tendency:
– Mode: the value in a string of numbers
that occurs most often
– Median: the value whose occurrence lies
in the middle of a set of ordered values
– Mean: sometimes referred to as the
―arithmetic mean‖; the average value
characterizing a set of numbers
6. Slide 6
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Understanding Data Via Descriptive Analysis
Measures of Variability:
– Frequency distribution reveals the number (percent)
of occurrences of each number or set of numbers
– Range identifies the maximum and minimum values
in a set of numbers
– Standard deviation indicates the degree of variation
in a way that can be translated into a bell-shaped
curve distribution
7. Slide 7
Descriptive statistics
If we wanted to characterize the students in a first year medical class we
would find that they are:
– Young
– Fit
– Male & Females
How young?
How fit is this class?
What is the distribution of males and females?
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8. Slide 8
Frequency distribution
The frequency with which observations
are assigned to each category or point
on a measurement scale.
– Most basic form of descriptive statistics
– May be expressed as a percentage of the total
sample found in each category
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9. Slide 9
Frequency distribution
The distribution is ―read‖ differently depending upon
the measurement level
– Nominal scales are read as discrete measurements at each level
– Ordinal measures show tendencies, but categories should not be
compared
– Interval and ratio scales allow for comparison among categories
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11. Slide 11
Grouped Relative Frequency Distribution
Relative Frequency Distribution of IQ for Two Classes
IQ FrequencyPercent Cumulative Percent
80 – 89 3 12.5 12.5
90 – 99 5 20.8 33.3
100 – 109 6 25.0 58.3
110 – 119 3 12.5 70.8
120 – 129 3 12.5 83.3
130 – 139 2 8.3 91.6
140 – 149 1 4.2 95.8
150 and over 1 4.2 100.0
Total 24 100.0 100.0
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12. Slide 12
Discrete and Continuous data
Data consisting of numerical (quantitative) variables can be further
divided into two groups: discrete and continuous.
1. If the set of all possible values, when pictured on the number line,
consists only of isolated points.
2. If the set of all values, when pictured on the number line, consists of
intervals.
The most common type of discrete variable we will encounter is a
counting variable.
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13. Slide 13
Accuracy & Precision
Accuracy: the closeness of the measurements to the “actual” or “real” value
of the physical quantity.
– Statistically this is estimated using the standard error of the mean
Precision: is used to indicate the closeness with which the measurements
agree with one another.
- Statistically the precision is estimated by the standard deviation of the
mean
Precision is related to random errors that can be dealt with using
statistics
Accuracy is related to systematic errors and are difficult to deal with
using statistics
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14. Slide 14
Accuracy and precision:
The target analogy
High accuracy but
low precision
High precision but
low accuracy
What does High accuracy and high precision look like?
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15. Slide 15
Accuracy and precision:
The target analogy
High accuracy and high precision
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16. Slide 16
Two types of error
Systematic error
– Poor accuracy
– Definite causes
– Reproducible
Random error
– Poor precision
– Non-specific causes
– Not reproducible
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17. Slide 17
Systematic error
Diagnosis
– Errors have consistent signs
– Errors have consistent magnitude
Treatment
– Calibration
– Correcting procedural flaws
– Checking with a different procedure
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18. Slide 18
Random error
Diagnosis
– Errors have random sign
– Small errors more likely than large errors
Treatment
– Take more measurements
– Improve technique
– Higher instrumental precision
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19. Slide 19
Statistical graphs of data
A picture is worth a thousand words!
Graphs for numerical data:
Histograms
Frequency polygons
Pie
Graphs for categorical data
Bar graphs
Pie
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20. Slide 20
Box-Plots
A way to graphically portray almost all the descriptive
statistics at once is the box-plot.
A box-plot shows: Upper and lower quartiles
Mean
Median
Range
Outliers (1.5 IQR)
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22. Slide 22
Bar Graphs
For categorical data
Like a histogram, but with gaps between bars to show
that each bar is a separate group.
Useful for showing two samples side-by-side
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23. Slide 23
Poor Below
Average
Average Above
Average
Excellent
Frequency
Rating
Bar Graph
1
2
3
4
5
6
7
8
9
10
Marada Inn Quality Ratings
Good?
Bad?
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24. Slide 24
Histograms
f on y axis (could also plot p or % )
X values (or midpoints of class intervals) on x axis
Plot each f with a bar, equal size, touching
No gaps between bars
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25. Slide 25
Pie Chart
The pie chart is a commonly used graphical device
for presenting relative frequency distributions for
qualitative data.
First draw a circle; then use the relative
frequencies to subdivide the circle
into sectors that correspond to the
relative frequency for each class.
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27. Slide 27
Skewness of distributions
Measures look at how lopsided distributions are—how far from the ideal
of the normal curve they are
When the median and the mean are different, the distribution is skewed.
The greater the difference, the greater the skew.
Distributions that trail away to the left are negatively skewed and those
that trail away to the right are positively skewed
If the skewness is extreme, the researcher should either transform the
data to make them better resemble a normal curve or else use a
different set of statistics—nonparametric statistics—to carry out the
analysis
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28. Slide 28
Symmetric
– Left tail is the mirror image of the right tail
– Examples: heights and weights of people
Histogram (Common categories)
RelativeFrequency
.05
.10
.15
.20
.25
.30
.35
0
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29. Slide 29
Histogram
Moderately Skewed Left
– A longer tail to the left
– Example: exam scores
RelativeFrequency
.05
.10
.15
.20
.25
.30
.35
0
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30. Slide 30
Moderately Right Skewed
– A Longer tail to the right
– Example: housing values
Histogram
RelativeFrequency
.05
.10
.15
.20
.25
.30
.35
0
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33. Slide 33
Symmetry: Kurtosis
A high kurtosis distribution has a sharper "peak"
and fatter "tails", while a low kurtosis
distribution has a more rounded peak with wider
"shoulders".
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35. Slide 35
correlations
determines whether and to what degree a
relationship exists between two or more
quantifiable variables
the degree of the relationship is expressed as a
coefficient of correlation.
the presence of a correlation does not indicate
a cause-effect relationship primarily because of
the possibility of multiple confounding factors
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36. Slide 36
…the presence of a correlation does
not indicate a cause-effect
relationship primarily because of the
possibility of multiple confounding
factors
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38. Slide 38
Spearman Rho...
…a measure of correlation used for
rank and ordinal data
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39. Slide 39
Pearson r...
…a measure of correlation used for
data of interval or ratio scales
…assumes that the relationship
between the variables being
correlated is linear
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40. Slide 40
So
Descriptive statistics are used to summarize data from
individual respondents, etc.
– They help to make sense of large numbers of individual
responses, to communicate the essence of those responses to
others
They focus on typical or average scores, the dispersion of
scores over the available responses, and the shape of the
response curve
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42. Slide 42
Used to:
Test for goodness of fit
Test for independence of attributes
Testing homogeneity
Testing given population variance
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43. Slide 43
Introduction
The test we use to measure the differences between
what is observed and what is expected according to
an assumed hypothesis is called the chi-square test.
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44. Slide 44
Important
The chi square test can only be used on
data that has the following characteristics:
The data must be in the
form of frequencies
The frequency data must have a
precise numerical value and must
be organised into categories or
groups.
The total number of observations
must be greater than 30.
The expected frequency in any one
cell of the table must be greater
than 5.
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45. Slide 45
Formula
χ 2 = ∑ (O – E)2
E
χ2 = The value of chi square
O = The observed value
E = The expected value
∑ (O – E)2 = all the values of (O – E) squared then
added together
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47. Slide 47
Construct a table with the information you have observed or
obtained.
Observed Frequencies (O)
Money Health Love Row
Total
men 82 446 355 883
women 46 574 273 893
Column total 128 1020 628 1776
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48. Slide 48
Work out the expected frequency.
Expected frequency = row total x column total
Grand total
money health love Row Total
men 63.63 507.128 312.23 883
women 64.36 512.87 315.76 893
Column Total 128 1020 628 1776
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49. Slide 49
For each of the cells calculate.
money health love Row
Total
Men 5.30 7.37 5.85
women 5023 7.29 5.8
Column Total χ2
Calc. =
36.873
(O – E)2
E
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50. Slide 50
χ2
Calc. = sum of all ( O-E)2/ E values in the cells.
Here χ 2
Calc. =36.873
Find χ 2
critical From the table with degree of freedom 2 and level of
significance 0.05
χ 2
Critical =5.99
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52. Slide 52
Conclusion
Compare χ2
Calc. and Χ2
critical obtained from the table
If χ2
Calc. Is larger than χ2
Critical. then reject null hypothesis and
accept the alternative
Here since χ 2
Calc. is much greater than χ 2
Critical, we can easily
reject null hypothesis
that is ; there lies a relation between the gender and choice of
selection.
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