3. Quartiles
Quartiles split the ranked data into 4 segments with
an equal number of values per segment
25% 25% 25% 25%
The first quartile, Q1, is the value for which 25% of the
observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are
larger)
Only 25% of the observations are greater than the third
quartile
Q1 Q2 Q3
4. Quartile Formulas
Find a quartile by determining the value in the
appropriate position in the ranked data, where
First quartile position: Q1 = n/4
Second quartile position: Q2 = n/2 (the median position)
Third quartile position: Q3 = 3n/4
where n is the number of observed values
5. (n = 9)
Q1 is in the 9/4 = 2.25 position of the ranked data
so Q1 = 12.25
Quartiles
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Example: Find the first quartile
Q1 and Q3 are measures of noncentral location
Q2 = median, a measure of central tendency
7. Same center,
different variation
Measures of Variation
Variation
Variance Standard
Deviation
Coefficient
of Variation
Range Interquartile
Range
Measures of variation give
information on the spread or
variability of the data
values.
8. Range
Simplest measure of variation
Difference between the largest and the smallest
values in a set of data:
Range = Xlargest – Xsmallest
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Range = 14 - 1 = 13
Example:
9. Ignores the way in which data are distributed
Sensitive to outliers
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
Disadvantages of the Range
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = 5 - 1 = 4
Range = 120 - 1 = 119
10. Coefficient of Range
Coefficient of Range = [L-S]/[L+S]
Where,
L = Largest value in the data set
S = Smallest value in the data set
11. Interquartile Range
Can eliminate some outlier problems by using
the interquartile range
Eliminate some high- and low-valued
observations and calculate the range from the
remaining values
Interquartile range = 3rd
quartile – 1st
quartile
= Q3 – Q1
15. Calculate the Coefficient of AD
(mean)
Sales (in thousand Rs) No. of days
10-20 3
20-30 6
30-40 11
40-50 3
50-60 2
16. Average (approximately) of squared deviations
of values from the mean
Sample variance:
Variance
1-n
)X(X
S
n
1i
2
i
2
∑=
−
=
Where = mean
n = sample size
Xi = ith
value of the variable X
X
17. Standard Deviation
Most commonly used measure of variation
Shows variation about the mean
Is the square root of the variance
Has the same units as the original data
Sample standard deviation:
1-n
)X(X
S
n
1i
2
i∑=
−
=
18. Calculation Example:
Sample Standard Deviation
Sample
Data (Xi) : 10 12 14 15 17 18 18 24
n = 8 Mean = X = 16
4.308
7
130
7
16)(2416)(1416)(1216)(10
1-n
)X(24)X(14)X(12)X(10
S
2222
2222
==
−++−+−+−
=
−++−+−+−
=
A measure of the “average”
scatter around the mean
26. Comparing Standard Deviations
Mean = 15.5
S = 3.33811 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5
S = 0.926
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
S = 4.567
Data C
28. Advantages of Variance and
Standard Deviation
Each value in the data set is used in the
calculation
Values far from the mean are given extra
weight
(because deviations from the mean are squared)
29. Coefficient of Variation
Measures relative variation
Always in percentage (%)
Shows variation relative to mean
Can be used to compare two or more sets of
data measured in different units
100%
X
S
CV ⋅
=
30. Comparing Coefficient
of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Standard deviation = $5
Both stocks
have the same
standard
deviation, but
stock B is less
variable relative
to its price
10%100%
$50
$5
100%
X
S
CVA =⋅=⋅
=
5%100%
$100
$5
100%
X
S
CVB =⋅=⋅
=
32. Calculate!
Calculate the coefficient of variation for
the following data set.
The price, in cents, of a stock over five trading days
was 52, 58, 55, 57, 59.
36. Z Scores
A measure of distance from the mean (for example, a
Z-score of 2.0 means that a value is 2.0 standard
deviations from the mean)
The difference between a value and the mean, divided
by the standard deviation
A Z score above 3.0 or below -3.0 is considered an
outlier
S
XX
Z
−
=
37. Z Scores
Example:
If the mean is 14.0 and the standard deviation is 3.0,
what is the Z score for the value 18.5?
The value 18.5 is 1.5 standard deviations above the
mean
(A negative Z-score would mean that a value is less
than the mean)
1.5
3.0
14.018.5
S
XX
Z =
−
=
−
=
(continued)
38. Shape of a Distribution
Describes how data are distributed
Measures of shape
Symmetric or skewed
Mean = MedianMean < Median Median < Mean
Right-SkewedLeft-Skewed Symmetric
39. If the data distribution is approximately
bell-shaped, then the interval:
contains about 68% of the values in
the population or the sample
The Empirical Rule
1σμ ±
μ
68%
1σμ ±
40. contains about 95% of the values in
the population or the sample
contains about 99.7% of the values
in the population or the sample
The Empirical Rule
2σμ ±
3σμ ±
3σμ ±
99.7%95%
2σμ ±
41.
42. Regardless of how the data are distributed,
at least (1 - 1/k2
) x 100% of the values will
fall within k standard deviations of the mean
(for k > 1)
Examples:
(1 - 1/12
) x 100% = 0% ……..... k=1 (μ ± 1σ)
(1 - 1/22
) x 100% = 75% …........ k=2 (μ ± 2σ)
(1 - 1/32
) x 100% = 89% ………. k=3 (μ ± 3σ)
Chebyshev Rule
withinAt least
43. Calculate!
Calculate the mean & the standard dev.
Calculate the percentage of
observations between the mean & 2σ±
Variable Frequency
44-46 3
46-48 24
48-50 27
50-52 21
52-54 5
44. Find the correct Mean & Std. dev.
The Mean & Standard Deviation of a set of 100
observations were worked out as 40 & 5
respectively. It was later learnt that a mistake had
been made in data entry – in place of 40 (the
correct value), 50 was entered.
45. Skewness
A fundamental task in many statistical analyses is to
characterize the location and variability of a data set
(Measures of central tendency vs. measures of
dispersion)
Both measures tell us nothing about the shape of the
distribution
It is possible to have frequency distributions which differ
widely in their nature and composition and yet may
have same central tendency and dispersion.
Therefore, a further characterization of the data
includes skewness
46. Positive & Negative Skew
Positive skewness
There are more observations below the mean than
above it
When the mean is greater than the median
Negative skewness
There are a small number of low observations and a
large number of high ones
When the median is greater than the mean
47. Measures of Skew
Skew is a measure of symmetry in the
distribution of scores
Positive
Skew
Negative Skew
Normal
(skew = 0)
49. The Rules
I. Rule One. If the mean is less than the median,
the data are skewed to the left.
II. Rule Two. If the mean is greater than the
median, the data are skewed to the right.
58. Calculate!
Obtain the limits of daily wages of the central 50% of the
workers
Calculate Bowley’s & Kelly’s Coefficients of Skewness
Wages No. of Workers
Below 200 10
200-250 25
250-300 145
300-350 220
350-400 70
Above 400 30