2-Descriptive statistics.pptx

Descriptive Statistics
Numerical Measures
Dr. Subhradev Sen
Alliance School of Business
ALLIANCE UNIVERSITY

Lecture Overview
• Measures of Location/Central Tendency
• Measures of Variability/Dispersion
• Measures of distribution shape, relative
location
• Weighted mean & Grouped Data
• Outliers & Box Plot
• Correlation Coefficient

MEASURES OF LOCATION
 Mean
 Median
 Mode
 Percentiles
 Quartiles

Median:
The median is another measure of central
location. The median is the value in the middle
when the data are arranged in ascending order
(smallest value to largest value).

Whenever a data set contains extreme values,
the median is often the preferred measure of
central location.

Mode:
The mode is the value that occurs with greatest
frequency.

Percentiles:
A percentile provides information about how
the data are spread over the interval from the
smallest value to the largest value.

Percentiles:
For data that do not contain numerous repeated
values, the pth percentile divides the data into
two parts. Approximately p percent of the
observations have values less than the pth
percentile; approximately (100-p) percent of the
observations have values greater than the pth
percentile.

Percentiles (Formal Definition):
The pth percentile is a value such that at least p
percent of the observations are less than or
equal to this value and at least (100-p) percent
of the observations are greater than or equal to
this value.

Percentiles:
Colleges and universities frequently report
admission test scores in terms of percentiles.

Percentiles:
Suppose an applicant obtains a raw score of 54 on
the verbal portion of an admission test. How this
student performed in relation to other students
taking the same test may not be readily apparent.
However, if the raw score of 54 corresponds to the
70th percentile, we know that approximately 70%
of the students scored lower than this individual
and approximately 30% of the students scored
higher than this individual.

Percentiles Calculation:

Percentiles Calculation Illustration:

Percentiles Calculation Illustration: Interest is to
find the 85th percentile.

Percentiles:
Note that the
50th percentile is the median.

Quartiles:
It is often desirable to divide data into four
parts, with each part containing approximately
one-fourth, or 25% of the observations.

Quartiles:

Quartiles: (Illustration)

Grouped Data: AVERAGE or MEAN
In case data are given as frequency distribution
or as grouped data. The mean is obtained by
Where

Example:
Value Frequency
1 5
2 9
3 12
4 17
5 14
6 10
7 6

Example:
Mean=299/73=4.09
Value(x) Frequency(f) xf
1 5 5
2 9 18
3 12 36
4 17 68
5 14 70
6 10 60
7 6 42
TOTAL 73=N 299

In case of grouped or continuous frequency
distribution, “x” is taken as the mid value of the
corresponding class

Example: Marks No. of Students
0-10 12
10-20 18
20-30 27
30-40 20
40-50 17
50-60 6

Example:
Mean=2800/100=28
Marks
No. of
Students
(f)
Mid-Point
(x) x.f
0-10 12 5 60
10-20 18 15 270
20-30 27 25 675
30-40 20 35 700
40-50 17 45 765
50-60 6 55 330
TOTAL 100 2800

Grouped Data: MEDIAN
Example:
Value Frequency
1 8
2 10
3 11
4 16
5 20
6 25
7 15
8 9
9 6

Example:
1. Find N/2, where N is the total frequency.
2. See the cumulative frequency just greater than
N/2.
3. The corresponding value of “x” is the median

N/2=120/2=60
Median=5
Value Frequency
Cumulative
frequency(<)
1 8 8
2 10 18
3 11 29
4 16 45
5 20 65
6 25 90
7 15 105
8 9 114
9 6 120

Wage No. of Employees
2000-3000 3
3000-4000 5
4000-5000 20
5000-6000 10
6000-7000 5

L: Lower limit of the median class.
f: Frequency of the median class.
h: Width of the median class.
C: Cumulative Frequency of the class just
preceding the median class.

N/2=43/2=21.5
Median = 4000+(1000/20)(21.5-8)
= 4000+675 = 4675
Wage
No. of
Employees cf
2000-3000 3 3
3000-4000 5 8
4000-5000 20 28
5000-6000 10 38
6000-7000 5 43

Grouped Data: MODE
Example:
Marks No. of Students
0-10 5
10-20 8
20-30 7
30-40 12
40-50 28
50-60 20
60-70 10
70-80 10

Grouped Data: MODE

Grouped Data: MODE
Marks
No. of
Students
0-10 5
10-20 8
20-30 7
30-40 12
40-50 28
50-60 20
60-70 10
70-80 10

MEASURES OF VARIABILITY
 Range
 Inter Quartile Range (IQR)
 Variance
 Standard Deviation
 Coefficient of Variation (CV)

Range:
Range is
Largest Value - Smallest Value

Range: (Illustration)
The largest starting salary is 3925 and the
smallest is 3310. The range is 3925-3310=615.

Inter Quartile Range (IQR):

Inter Quartile Range (IQR): (Illustration)

Variance:
For Raw Data:
For Frequency Data:
Sample variance is denoted by 𝒔𝟐
(Not by σ2
),
σ2
is used for population variance indication.

Variance:

Standard Deviation:
The standard deviation (sd) is defined to be the
positive square root of the variance. It is
denoted by σ.
Sample sd is denoted by s
For the previous example,

Note:
Note that when we deal with sample, standard
deviation notation is “s”.

Coefficient of Variation (CV):
In some situations we may be interested in a
descriptive statistic that indicates how large the
standard deviation is relative to the mean.
This measure is called the coefficient of
variation and is usually expressed as a
percentage.

Coefficient of Variation (CV):
CV is expressed in percentage and is utilized for
comparing performances of individual on a similar
variable of measurement

Moments:
 Raw Moments (Moments about origin “0”)
 Central Moments (Moments about sample
mean)

Moments: Raw Moments (Moments about
origin “0”)

Moments: Central Moments (Moments about
mean)

MEASURES OF DISTRIBUTION SHAPE
Measures of Skewness:

Other Measures of Skewness:

Pooled Mean & Pooled Variance
Suppose the data is given in different groups in
the following fashion.
Groups Sample Size Sample
Mean
Sample
Variance
Group-1 𝐧𝟏 𝐱𝟏 𝐬𝟏
𝟐
Group-2 𝐧𝟐 𝐱𝟐 𝐬𝟐
𝟐
Group-3 𝐧𝟑 𝐱𝟑 𝐬𝟑
𝟐

Pooled Mean & pooled Variance
Then to find
overall mean taking all the groups together we
use the concept of pooled mean or Overall
Mean.
overall variance taking all the groups together
we use the concept of pooled variance or
Composite variance.

Pooled Mean & Pooled Variance
Pooled Mean:
Pooled Variance (or Composite Variance):
where

MEASURES OF RELATIVE LOCATION
Z-Score:
Measures of relative location help us determine
how far a particular value is from the mean.

Z-Score:
Z-score is called a standardized value

Z-Score:
The z-score, 𝑧𝑖 , can be interpreted as the
number of standard deviations 𝑥𝑖 is from the
mean .
For example, 𝑧1= 1.2 would indicate that 𝑥1 is
1.2 standard deviations greater than the sample
mean.
Similarly, 𝑧2= - 0.5 would indicate that 𝑥2 is .5,
or 1/2, standard deviation less than the sample
mean.

Data Exploration
Five Number Summary:
In a five-number summary, the following five
numbers are used to summarize the data:

Data Exploration
Five Number Summary:
For Graduate Salary Data:

Data Exploration
Box Plot:
A box plot is a graphical summary of data that is
based on a five-number summary.

Data Exploration
Outliers:
Sometimes a data set will have one or more
observations with unusually large or unusually
small values. These extreme values are called
outliers.

Data Exploration
Outliers:
Experienced statisticians take steps to identify
outliers and then review each one carefully.

Data Exploration
Outliers:
 An outlier may be a data value that has
been incorrectly recorded. If so, it can be
corrected before further analysis.
 An outlier may also be from an observation
that was incorrectly included in the data
set; if so, it can be removed.

Data Exploration
Outliers:
 Finally, an outlier may be an unusual data
value that has been recorded correctly and
belongs in the data set. In such cases it
should remain.

Correlation
Sometime we may be interested in finding out
relationship measure between two variables
considered in our data.

Correlation
Covariance between X and Y

Correlation
Covariance:
Example: Sample data for Stereo & Sound
Equipment.

Correlation
Covariance:
Scatter diagram

Correlation
Interpretation of Covariance:

Pearson’s Correlation Coefficient
Karl Pearson’s Correlation Coefficient:
A unit free measure which measures the linear
relationship between two variables.

Where,

Pearson’s Correlation Coefficient:

Pearson’s Correlation Coefficient:
We conclude that a strong positive linear
relationship occurs between the number of
commercials and sales.
More specifically, an increase in the number of
commercials is associated with an increase in
sales.

Pearson’s Correlation Coefficient: Note
1. rxy ranges from -1 to +1
2. It measures linear relationship between variables
3. Need quantitative variables to calculate the
coefficient.
4. Correlation Coefficient is unit free.

Home Work: (1)

2-Descriptive statistics.pptx

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