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1PPTs-Handout One-An Overview of Descriptive Statistics-Chapter 1_2.pptx
1. Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk
2. Brief Introduction
• I am Dr. Abdul Aziz, Assistant Professor and HOD Business
Administration Department at Federal Urdu University Karachi.
• Having more than 15 years of teaching experience.
• Having good command on statistics, statistical inferences,
econometrics, quantitative research methods, multivariate data
analysis, corporate finance, financial management, strategic
finance, financial engineering, and financial derivatives subjects.
• Also having good command to analyze questionnaire data, time
series data, and panel data by using SPSS, eviews and STATA
softwares.
• So far, seventeen students have successfully completed their MS
under my supervision and Twenty two research papers
published in different national and international journals till 20-
03-2021. Four papers have recently been submitted for
publication .
• Have worked on diversified topics and used advanced
econometrics techniques such as ARCH family models, co-
integration, VECM, VAR models, and non parametric tests
specially used for ordinal and nominal scales data.
Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk
3. Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk
4. An Overview of Descriptive Statistics
Handout One-Chapter-1 & 2
(Practice Problems)
Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk
5. Q-1: Specify the mathematical symbol used for each of
the following descriptive measures.
A. Sample mean B. Sample standard deviation
C. Population mean D. Population stand. deviation
Answer1:
A. Symbol of sample mean
A. Symbol of sample standard deviation S
A. Symbol of population mean µ
A. Symbol of population standard deviation σ
Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk
6. Q-2: How is a z-score obtained? What is the interpretation of a z-
score? An observation has a z-score of 2.9. Roughly speaking, what is
the relative standing of the observation?
Answer2:
Formula to obtain z-score
Z-score is the standard normal score of particular x value. It is also called measure of
relative standing
approximately 3 times stranded deviation
away from mean. See graph for further
details.
If z-score of an observation is the 2.9.
It means particular observation deviates
Dr. Abdul Aziz, Ph.D. in Business
Administration from University of Sindh.
abdul.aziz@fuuast.edu.pk
2.9
7. Q-3: Given the following index numbers for 10 years. Calculate
arithmetic, geometric, harmonic means.
Answer3:
101 99 95 105 102 87 85 112 110 112
8. 3 1 2 0 4 6 1 0 1 5 2 3 3 0 7 2 2 0 3 1 2 2 4 2 4 0 4 3 1 4
Q-4: The number of children in each of the thirty families is as follows.
Calculate median, Quartile1, Quartile3, Decile1, percentile60, mean and mode.
Answer4:
Median is the middle value (At 50% position) of data. First arrange data in ascending order
Median Position = or 0.5(n+1) There is no exact 15.5 position in
data, therefore we take average of 15th and 16th position numbers. At 15th position number is
2 and at 16th position number is also 2, average of both numbers is 2 and that is our median.
Median means 50% of the observations lie at the left side of 2 and 50% observations lie at
the right side of 2.
Quartile1 is the 25th percent value of the data. First arrange data in ascending order
Quartile1 Position= or 0.25(n+1) . There is no exact 7.75 position in
data,
therefore we take average of 7th and 8th positions numbers. The average is 1, so Q1 = 1. Q1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 6 7
50%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 6 7
25%
9. 3 1 2 0 4 6 1 0 1 5 2 3 3 0 7 2 2 0 3 1 2 2 4 2 4 0 4 3 1 4
Answer4:
Quartile3 is the 75th percent value of the data.
or 0.25(3n+1) = 22.75. Q3 is the average 22nd and 23rd
which 3.5. It indicates 75% of the observations lie at the left side of 3.5 and 25% of the
observations lie at the right side of 3.5. Similarly D1 is the 10th percent value, P60 is the 60th
percent value. Remember Median = Q2 = D5 = P50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 6 7
75%
Q-4: The number of children in each of the thirty families is as follows.
Calculate median, quartile1, quartile3, decile1, percentile60, mean and mode.
X f
0 5
1 5
2 7
3 5
4 5
5 1
6 1
7 1
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7
F
X
Data is summarized in table and a
graph is developed. This step shows
whether data is normally distributed
or non normal. This graph is made to
develop the basics about skewness
and kurtosis concepts that will be
discussed in Q-18.
Mode is the most repetitive number in the
data i.e. 2 because it is the most repetitive
number in the data occurring seven times.
10. Q-5: For the data given below, calculate arithmetic, geometric,
harmonic means, median, mode, S, mean deviation (M.D), S2,
Coefficient of variation CoV, range, quartile deviation and
interquartile range. Solve Q-6 and Q-7, refer Q-1 to Q-5.
2 7 5 -4 0 3 11 9 -2 0 -1 4 0 -2 5
Answer5:
For arithmetic, geometric, and harmonic means use following formulas:
jfjhaiaiman bano game
Note: Geometric mean can not be calculated because of negative observations.
For median and mode calculations, refer Q-4.
Range = Maximum-Minimum = 11- (-4) =15
Interquartile Range =Q3 – Q1 = 5 – (-1) = 6
11. Q-8: For the following frequency distribution, calculate
arithmetic, geometric, harmonic means, median, mode, S, (M.D),
S2, CoV, quartile deviation, interquartile range and its lower,
upper limits.
Answer8:
Class Boundaries 10-12 12-14 14-16 16-18 18-20
Frequency 14 26 42 30 8
12. Q-8: For the following frequency distribution, calculate
arithmetic, geometric, harmonic means, median, mode, S, (M.D),
S2, CoV, quartile deviation, interquartile range and its lower,
upper limits. (Continued)
Answer8:
Class Boundaries 10-12 12-14 14-16 16-18 18-20
Frequency 14 26 42 30 8
Class Boundaries f Midpoint=
x
Log (x) fLog(x)
10-12 14 11 1.041393 14.58 1.273
12-14 26 13 1.113943 28.96 2
14-16 42 15 1.176091 49.4 2.8
16-18 30 17 1.230449 36.91 1.765
18-20 8 19 1.278754 10.23 0.421
∑f=n=120 ∑fLog(x)=140.1 8.258
13. Q-8: For the following frequency distribution, calculate
arithmetic, geometric, harmonic means, median, mode, S, (M.D),
S2, CoV, quartile deviation, interquartile range and its lower,
upper limits. (Continued)
Answer8: Find median class by dividing i.e. , Highlight near largest class.
Find Quartile1 class dividing i.e. , Highlight near largest class.
Find Quartile3 class by dividing i.e. , Highlight near largest class.
Class Boundaries 10-12 12-14 14-16 16-18 18-20
Frequency 14 26 42 30 8
Class
Boundaries
f Cumulative
f
Class
Boundaries
f Cumulative f
10-12 14 14 10-12 14 14 C
12-14 26 40 C l=12 12-14 h=2 26 f 40 Near largest
l=14 14-16 h=2 42 f 82 Near largest 14-16 42 82 C
16-18 30 112 l=16 16-18 h=2 30 f 112 Near largest
18-20 8 120 18-20 8 120
∑f=n=120 ∑f=n=120
14. Q-8: For the following frequency distribution, calculate
arithmetic, geometric, harmonic means, median, mode, S, (M.D),
S2, CoV, quartile deviation, interquartile range and its lower,
upper limits. Solve Q-9 to Q-14, refer Q-8.
Answer8:
Class Boundaries 10-12 12-14 14-16 16-18 18-20
Frequency 14 26 42 30 8
Class Boundaries f
10-12 14
12-14 26 f1
l=14 14-16 h=2 42 fm
16-18 30 f2
18-20 8
∑f=n=120
Interquartile Range = Q3 – Q1= 16.53 – 13.08 = 3.45),
Lower limit = Q1 - 1.5 x IQR, Upper limit= Q3 + 1.5 x IQR
Lower limit = 13.08 – 1.5 x 3.45, Upper limit = 16.53 + 1.5 x 3.45
Lower limit = 7.905 and Upper limit = 21.705
15. Q-15: For the following frequency distribution. Calculate four
moments of origin and convert them into moments about mean.
Also calculate skewness and kurtosis.
Answer15:
Class Boundaries f Cumulative f Midpoint=x fx fx2 Fx3
fx4
-0.5 – 2.5 4 4 1 4 4 4 4
2.5 – 5.5 7 11 4 28 112 448 1792
5.5 – 8.5 7 18 C 7 49 343 2401 16807
l=8.5 8.5 – 11.5 h=3 15 f 33 near largest 10 150 1500 15000 150000
11.5 – 14.5 10 43 13 130 1690 21970 285610
14.5 – 17.5 6 49 16 96 1536 24576 393216
17.5 – 20.5 1 50 19 19 361 6859 130321
50 476 5546 71258 977750
Class Interval 0-2 3-5 6-8 9-11 12-14 15-17 18-20
Frequency 4 7 7 15 10 6 1
16. Q-15: For the following frequency distribution. Calculate four
moments of origin and convert them into moments about mean.
Also calculate skewness and kurtosis.
Answer15:
Class Boundaries f Midpoint=x
-0.5 – 2.5 4 1 -34.08 290.3616 -2473.88083 21077.46
2.5 – 5.5 7 4 -38.64 213.2928 -1177.37626 6499.117
5.5 – 8.5 7 7 -17.64 44.4528 -112.021056 282.2931
8.5 – 11.5 15 10 7.2 3.456 1.65888 0.796262
11.5 – 14.5 10 13 34.8 121.104 421.44192 1466.618
14.5 – 17.5 6 16 38.88 251.9424 1632.58675 10579.16
17.5 – 20.5 1 19 9.48 89.8704 851.971392 8076.689
50 0 1014.48 -855.6192 47982.14
Class Interval 0-2 3-5 6-8 9-11 12-14 15-17 18-20
Frequency 4 7 7 15 10 6 1
17. Q-15: For the following frequency distribution. Calculate four moments of origin and
convert them into moments about mean. Also calculate skewness and kurtosis.
Solve Q-16 and Q-17 (For Q-17, refer formulas given in last slide). Refer Q-15
Answer15: Another way to calculate moments about mean by using moment of origin:
Class Interval 0-2 3-5 6-8 9-11 12-14 15-17 18-20
Frequency 4 7 7 15 10 6 1
18. Q-18: What is the skewness? What is symmetrical, right skewed and left
skewed distribution and its properties? What is the kurtosis? What is
leptokurtic, mesokurtic and platykurtic distribution and its properties?
Answer18:
Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A
distribution, or data set, is symmetric if it looks the same to the left and right of the
center point.
Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to
a normal distribution. That is, data sets with high kurtosis tend to have heavy tails,
or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers. A
uniform distribution would be the extreme case.
19. Q-18: What is the skewness? What is symmetrical, right skewed and left
skewed distribution and its properties? What is the kurtosis? What is
leptokurtic, mesokurtic and platykurtic distribution and its properties?
Answer18:
20. Q-19: First two moment about origin of a frequency
distribution are given as =26 and =932. Find the mean
and standard deviation of the distribution.
Answer19:
As we discussed Q-15, first order of moment about origin is mean so mean of the
data is 26. We also checked in Q-15 that second order of moment about mean is
variance. The standard deviation is the square root of the variance, first calculate
second order moments about mean by using two moments about origin then
calculate standard deviation.
Q-20: The mean, median and S.D of a frequency distribution
are computed as = 27.2, median = 24.6 and S.D = 16. Find
the coefficient of skewness of the distribution. Is it positive
or negative skewed?
Answer20: Its coefficient is 0.4875 which is positively skewed. This data has mean > median > mode.
21. Q-21: The three quartiles of a frequency distribution are
given as: Q1 = 43.2, Q2 = 50.5 and Q3 = 57.8. Comment on
the symmetry of the distribution.
Answer21:
Solve Q-22 to Q-26.
For Q-22, refer Q-21
For Q-23, refer Q-15
For Q-24, refer Q-8 and Q-15.
For Q-24, use following formula for calculations.
Same have been used to solve Q-17.
For Q-26, refer Q-3 and Q-15.
