3. Ratio
• is a relationship between two numbers of the
same
kind (e.g., objects, persons, students, spoonful
s, units of whatever identical dimension)
• usually expressed as "a to b" or a:b
• For example, suppose I have 10 pairs of socks
for every pair of shoes then the ratio of
shoes:socks would be 1:10 and the ratio of
socks:shoes would be 10:1
2/4/2012 Dr. Riaz A. Bhutto 3
4. Coefficient of variation
• Is the ratio of the standard deviation to the
mean
• Used to compare the relative variation or spread
of the distribution of different series, samples, or
population; or the different characteristics of a
single series
• Expressed in percentage
SD
CV (%) = -------- X 100
__
X
2/4/2012 Dr. Riaz A. Bhutto 4
5. Example
• In a medical college, the mean weight of 100
medical students is 140 lbs, with S.D of 28 lbs.
The mean height of these students is 66”, with
S.D of 6”
• CV for weight = 28/140 x 100 = 20%
CV for height = 6/66 x 100 = 9%
• Based on the CV, therefore, the relative spread
of weight among the students is greater than
that of height
2/4/2012 Dr. Riaz A. Bhutto 5
6. Percentile
• Are the points which divide all
measurements/value into 100 equal parts
• In statistics, a percentile (or centile) is the
value of a variable below which a
certain percent of observations fall.
• For example, the 20th percentile is the value
(or score) below which 20 percent of the
observations may be found.
2/4/2012 Dr. Riaz A. Bhutto 6
7. • The 25th percentile is also known as the
first quartile (Q1), the 50th percentile as
the median or second quartile (Q2), and the
75th percentile as the third quartile (Q3).
2/4/2012 Dr. Riaz A. Bhutto 7
8. Normal Distribution
• Data can be "distributed" (spread out) in
different ways.
• It can be spread out more on the left ... or
more on the right
• Or it can be all jumbled up
2/4/2012 Dr. Riaz A. Bhutto 8
9. A Normal Distribution
But there are many cases where the data tends to be around a central value with
no bias left or right, and it gets close to a "Normal Distribution" like this:
The "Bell Curve" is a Normal Distribution.
It is often called a "Bell Curve"
because it looks like a bell.
2/4/2012 Dr. Riaz A. Bhutto 9
10. Many things closely follow a Normal Distribution:
Heights of people
Size of things produced by machines
Errors in measurements
Blood pressure
Marks on a test
We say the data is "normally distributed".
2/4/2012 Dr. Riaz A. Bhutto 10
11. The Normal Distribution has:
mean = median = mode
symmetry about the center
50% of values less than the mean
and 50% greater than the mean
2/4/2012 Dr. Riaz A. Bhutto 11
12. 68% of values are within
1 standard deviation of the mean
95% are within 2 standard deviations
99.7% are within 3 standard deviations
2/4/2012 Dr. Riaz A. Bhutto 12
13. Example: 95% of students at school are between 1.1m and 1.7m tall.
Assuming this data is normally distributed can you calculate the mean and standard
deviation?
The mean is halfway between 1.1m and 1.7m:
Mean = (1.1m + 1.7m) / 2 = 1.4m
95% is 2 standard
deviations either side
of the mean (a total of
4 standard deviations)
so:
1 standard deviation
= (1.7m-1.1m) / 4
= 0.6m / 4 = 0.15m
And this is the result:
It is good to know the standard deviation, because we can say that any value is:
likely to be within 1 standard deviation (68 out of 100 will be)
very likely to be within 2 standard deviations (95 out of 100 will be)
almost certainly within 3 standard Riaz A. Bhutto (997 out of 1000 will be)
2/4/2012 Dr. deviations 13
14. Standard Scores
The number of standard deviations from the
mean is also called the "Standard
Score", "sigma" or "z-score". Get used to
those words!
Example: In that same school one of your friends is 1.85m tall
You can see on the bell curve that 1.85m is 3
standard deviations from the mean of 1.4, so:
Your friend's height has a "z-score" of 3.0
It is also possible to calculate how many
standard deviations 1.85 is from the mean
How far is 1.85 from the mean?
It is 1.85 - 1.4 = 0.45m from the mean
How many standard deviations is that? The
standard deviation is 0.15m, so:
0.45m / 0.15m = 3 standard deviations
2/4/2012 Dr. Riaz A. Bhutto 14
15. So to convert a value to a Standard Score ("z-score"):
•first subtract the mean,
•then divide by the Standard Deviation
And doing that is called "Standardizing":
You can take any Normal Distribution and convert it to The
Standard Normal Distribution.
2/4/2012 Dr. Riaz A. Bhutto 15
17. Tables
• Simple tables
• Frequency Distribution table
• Cumulative Frequency table
• Relative Frequency table
2/4/2012 Dr. Riaz A. Bhutto 17
18. Frequency Distribution
Rating Frequency
Poor 2
Below Average 3
Average 5
Above Average 9
Excellent 1
Total 20
19. Relative Frequency and
Percent Frequency Distributions
Relative Percent
Rating Frequency Frequency
Poor .10 10
Below Average .15 15
Average .25 25 .10(100) = 10
Above Average .45 45
Excellent .05 5
Total 1.00 100
1/20 = .05
20. Tabulating Numerical Data:
Cumulative Frequency
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Cumulative Cumulative
Class Frequency % Frequency
10 but under 20 3 15
20 but under 30 9 45
30 but under 40 14 70
40 but under 50 18 90
50 but under 60 20 100
21. Tabulating Numerical Data: Frequency
Distributions
(continued)
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Relative
Class Frequency Frequency Percentage
10 but under 20 3 .15 15
20 but under 30 6 .30 30
30 but under 40 5 .25 25
40 but under 50 4 .20 20
50 but under 60 2 .10 10
Total 20 1 100
22. Charts and Graphs
Bar Charts (for presentation of categorical
data)
• Simple
• Multiple
• Component
Pie Charts / graph (for presentation of
categorical data)
Dot frequency graphs
2/4/2012 Dr. Riaz A. Bhutto 22
23. Good?
Bar Graph
Bad?
Quality Ratings
10
9
8
7
Frequency
6
5
4
3
2
1
Rating
Poor Below Average Above Excellent
Average Average
26. Pie Chart
Most common way of presenting the categorical data. The value of each category is
divided by the 360° and then each category is allocated the respective angles to present the
proportion it has.
2/4/2012 Dr. Riaz A. Bhutto 26
27. Dot Frequency Plot
Tune-up Parts Cost
.
. .. . . .
. . .. ..... .......... .. . .. . . ... . ... .
.. .. .. .. .
50 60 70 80 90 100 110
Cost (Rs.)
Not used much anymore. Common when
graphical drawing tools were primitive.
28. Diagrams
• Histogram (for presentation of continuous
data)
• Frequency polygon
• Line diagram
• Pictogram (are a form of bar charts)
• Scatter diagram (shows the relationship
between two variables)
2/4/2012 Dr. Riaz A. Bhutto 28
29. Histogram
Tune-up Parts Cost
18
16
14
12
Frequency
10
8
6
4
2
Parts
50 59 60 69 70 79 80 89 90 99 100-110 Cost (Rs.)
