1) Skewness is a measure of the lack of symmetry in a distribution. It indicates if the distribution is balanced around the mean. 2) There are several ways to calculate skewness including using the mean, median, mode, quartiles, or moments. Positive skewness means the mean is greater than the median, while negative skewness means the mean is less than the median. 3) Examples show how to calculate skewness using different formulas and interpret if a distribution is positively, negatively, or symmetrically skewed based on the resulting skewness value.

1. NADEEM UDDIN
ASSOCIATE PROFESSOR
OF STATISTICS

2. Skewness:
Skewness is a measure of departure 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.
Measures of skewness are.
MEAN MODE
1) Pearson`s first Coefficient of Skewness = . .C S K
S.D


 3 MEAN MEDIAN
2) Pearson`s Second Coefficient of Skewness = . .C S K
S.D


3 1 2
3 1
2
3) Bowley`s Quartile Coefficient of Skewness = . .
Q Q Q
C S K
Q Q
 


4) Measure of Skewness = 𝛽1 =
𝜇3
2
𝜇2
3

3. Characteristics of a symmetrical distribution
1- The mean, median and mode are same.
2- The upper and lower quartile are equidistance from median.
𝑖. 𝑒. 𝑄3 − 𝑚𝑒𝑑𝑖𝑎𝑛 = 𝑚𝑒𝑑𝑖𝑎𝑛 − 𝑄1
3- The sum of deviations from median of any series is zero.
𝑖. 𝑒. 𝑥 − 𝑚𝑒𝑑𝑖𝑎𝑛 = 0
4- All the odd ordered moments about mean are zero.

4. Characteristics of a positively skewed distribution
1- Mean ˃ median ˃ mode .
2- The upper and lower quartile are equidistance from median.
𝑖. 𝑒. 𝑄3 − 𝑚𝑒𝑑𝑖𝑎𝑛 ˃ 𝑚𝑒𝑑𝑖𝑎𝑛 − 𝑄1
3- The sum of deviations from median of any series is greater than
zero.
𝑖. 𝑒. 𝑥 − 𝑚𝑒𝑑𝑖𝑎𝑛 ˃ 0
4- All the odd ordered moments about mean are greater than zero.

5. Characteristics of a negatively skewed distribution
1- Mean ˂ median ˂ mode .
2- The upper and lower quartile are equidistance from median.
𝑖. 𝑒. 𝑄3 − 𝑚𝑒𝑑𝑖𝑎𝑛 ˂ 𝑚𝑒𝑑𝑖𝑎𝑛 − 𝑄1
3- The sum of deviations from median of any series is less than
zero.
𝑖. 𝑒. 𝑥 − 𝑚𝑒𝑑𝑖𝑎𝑛 ˂ 0
4- All the odd ordered moments about mean are less than zero.

6. The concept of symmetry, positive skewness and negative
skewness can be define by the following figure.
Symmetrical curve

8. Interpretation of result:
Skewness lies between -1 to +1.
If the skewness result is zero then the distribution is symmetrical.
If the skewness result is positive then the distribution is positively skewed.
If the skewness result is negative then the distribution is negatively skewed.

9. Example-1:
If Mean=28, Mode=23 and S.d=4.2 find the coefficient of skewness.
Solution:
S.D
KSC
MODEMEAN
..


28 23
1.19
4.2
coefficient of skewness

 
Comments
Distribution is positively skewed.

10. Example-2:
If Mean=75.3, Mode=84 and S.d=12.06 find the coefficient of skewness.
Solution:
S.D
KSC
MODEMEAN
..


75.3 84
0.72
12.06
coefficient of skewness

  
Comments
Distribution is negatively skewed.

11.  
S.D
KSC
MEDIANMEAN3
..


 3 13 13
0
5.83
coefficient of skewness

 
Example-3:
If Mean=13, Median=13 and S.d=5.83 find the coefficient of skewness.
Solution:
Comments
Distribution is symmetrical.

12. 13
213 2
..
QQ
QQQ
KSC



 60.46 49.65 2 54.79
0.05
60.46 49.65
coefficient of skewness
 
 

Example-4:
If Q1 = 49.65, Q2 = 54.79 and Q3 = 60.46 find the coefficient of skewness.
Solution:
Comments
Distribution is positively skewed.

13. Example-5:
If 𝜇2 = 15 , and 𝜇3 = 39 find the coefficient of skewness.
Solution:
𝛽1 =
𝜇3
2
𝜇2
3
𝛽1 =
39 2
15 3
𝛽1 = 0.4507
Comments
Distribution is positively skewed.

14. Example-6:
If 𝜇2 = 16 , and 𝜇3 = −64 find the coefficient of skewness.
Solution:
𝛽1 =
𝜇3
2
𝜇2
3
𝛽1 =
−64 2
16 3
𝛽1 = 1
Comments
Distribution is negatively skewed (Because 𝜇3 is negative).