Measures of Variation,Skewness, & Kurtosis

My fellow classmates in MBA I'm Jamaica Austria here is your copy of Group #5 report in Business Statistics & I'm with my co-reporter Sir Norvadez Gonzales.


Chapter 5: Measures of Variation, Skewness, and Kurtosis:

Importance of measures of variation:

- Measures of variation are statistics that indicate the degree to which numerical data tend to spread about an average value. It is also called dispersion, scatter, spread.
- In the simple words “the measurement of the scattering of the mass of figures (data) in a series about an average is called measure of variation”.

Measures of Variation:

1. Range
2. Quartile Deviation
3. Average or Mean Deviation
4. Standard Deviation
5. Coefficient of Variation
6. Standard Score

Range:

The Range is the simplest measure of variation. It is defined simply as thedifference between the highest value and the lowest value of observation in aset of data.

Note: These formulas applicable for grouped and ungrouped data.

Range = Highest Value – Lowest Value

Coefficient of Range = highest value - lowest value/highest value + lowest value

Quartile Deviation:

Quartile Deviation is another measure of variation, also termed as semi-inter
quartile range.

Qd = Q3 - Q1 / 2

Note: Where in asymmetrical distribution are not equidistant from median.

Coefficient of Quartile deviation or the relative measure of Qd:

Coefficient of Qd = Q3 – Q1 / Q3 + Q1

Note: These formulas applicable for grouped and ungrouped data.

Q1 = L1 + { [ (n/4-cf) / f ] (Xi) }

Where: n= total sum of observation
Xi=class interval (for the daily expenditure of students)
f=frequency ( for the number of students)
cf=cumulative frequency
L1=lower limit of class intervals


Q3 = L1 + { [ (3n/4-cf) / f ] (Xi) }

Where: n= total sum of observation
Xi=class interval
f=frequency
cf=cumulative frequency
L1=lower limit of class intervals


Average or Mean Deviation:

- the mean deviation overcomes the above weakness by considering all the items of a data set. The mean deviation is the arithmetic mean of absolute difference between the items in a distribution and the average of that distribution. Theoretically, mean deviation can be computed from the mean or the median or the mode. However, in actual practice the mean is frequently used in computing the mean deviation. Under this method, algebraic signs (+, –) are ignored while taking the deviations from average.

For grouped data, the formula is:

Md = ∑ f X - X / N

For ungrouped data, the formula is:

Md = ∑ X - X / N

Where:
Md = Mean deviation

X- X = absolute value of the difference between each items and the mean.
It is read: the absolute value of X minus X bar.
N = number of items

For Coefficient of Mean (Md):

Coefficient of Md = Md / Mean


Standard Deviation:

- Standard deviation is the most familiar, important and widely used measure of variation. It is a significant measure for making comparison of variability between two or more sets of data in terms of their distance from the mean.

For ungrouped data, the formula is:

σ = ∑ (X-X) / N-1

For grouped data, the formula is:

σ = ∑ f(X-X) / N-1

where:

σ = standard deviation

(X-X) = the deviation of each value of the series from the mean
N = number of observation

Coefficient of Variation:

- The coefficient of variation expresses the standard deviation as a percentage of the mean. Expressed in percent, it can be used to compare the variability of two or more distributions even when the observations are expressed in different units of measurement.

Interpretation:
- When coefficient of variable is lesser in the data, it is said to be more consistent or have less variability.
- On the other hand, the series having higher coefficient of variable has higher degree of variability or lesser consistency.
- And it is said not satisfactory useful when the mean is close to zero.

Vc = σ / X (100)

Where:

Vc = Coefficient of varaiation
σ = standard deviation
_
X = Mean

Standard Score:

- Another measure of relative variation is the standard score. It tells the relative location of a particular raw score with regard to the mean of all the scores in a series.
- Raw score are scores or measurements which are expressed in the form in which they were originally obtained, like test score, weight, and height measurements.

It can be expressed to:

z = X - X / σ

Where:

z = standard score
X = given raw score
_
X =mean
σ = standard deviation

It can also be expressed to:

t = 10z + 50

where:

t = another type of standard score (if z or raw score is already given)


SKEWNESS & KURTOSIS:

Uses of Skewness and Kurtosis

- The measures of skewness and kurtosis indicate the extent of departure of a distribution from normality and permit comparison of two or more distributions.


SKEWNESS:

- the terms “skewed” and “askew” are used to refer to something that is out of line or distorted on one side.
- measures the degree of symmetry in a frequency distribution by determining the extent to which the values are evenly or unevenly distributed on either side of the mean.

Interpretation:

- If Sk=0, the distribution is a normal curve.
- If Sk > 0, the distribution is more peaked than the normal curve.
- If Sk < 0, the distribution is less peaked than the normal curve.

- Symmetry signifies that the values of variable are equidistant from the average on both sides. In other words, a balanced pattern of a distribution is called symmetrical distribution, where as unbalanced pattern of distribution is called asymmetrical distribution.

There are two important methods for measuring the coefficient of skewness.

1. Karl Pearson’s coefficient of skewness.
2. Bowley's coefficient of skewness.

1. Karl Pearson’s Coefficient of Skewness: (denoted as SKp.)
- This co-efficient of skewness, is obtained by dividing the difference between the mean and the mode by the standard deviation.

The formula of Pearson’s coefficient of skewness denoted as SKp can be expressed in terms of:

Skp= X-Mo/σ

Pearson’s alternative formula for coefficient of skewness:

SKp= 3(Mean-Median)/σ

2. Bowley's Measure of Co-efficient of Skewness
- Bowley's method for coefficient of skewness (SKb) is derived from quartile values and for this reason it is useful in case of open-ended distribution, whereextreme values are presented and/or class intervals are unequal in the collecteddata or the median and quartile values only are available.
The formula ofBowley’s coefficient of skewness denoted as SKb can be expressed in terms of:

SKb = (Q3-Q2) - (Q2-Q1) / (Q3-Q2) + (Q2-Q1)

Alternatively,

SKb=Q3 + Q1- 2 Median/Q3 - Q1

KURTOSIS:

Kurtosis – measures the flatness or peakness of a frequency distribution.

Interpretation:

- If ku=3, the distribution is mesokurtic or it is a normal curve.
- If ku > 3, the distribution is leptokurtic or more peaked than the normal curve.
- If ku <3, the distribution is platykurtic or less peaked than the normal curve.

The moment coefficient of Kurtosis can be expressed in terms of:


Where: a4 = moment coefficient of kurtosis
m4 = fourth moment
_
X = mean
m2 = second moment (variance)
square root of m2 = standard deviation
n = number of data points

End of our Report