1. Module - 58
Descriptive Statistics
Basics
CMT LEVEL - I

2. Descriptive statistics
• Descriptive statistics are brief descriptive coefficients that
summarize a given data set, which can be either a
representation of the entire or a sample of a population.
• Descriptive statistics are broken down into measures of central
tendency and measures of variability (spread).
• Measures of central tendency include the mean, median, and
mode
• while measures of variability include the standard deviation,
variance, the minimum and maximum variables, and the
kurtosis and skewness.

3. Measures of Central Tendency
Mean Median Mode
Geometric
Mean
Weighted
Arithmetic
Mean

4. Mean (or average)
•The mean (or average) is the most popular and well known
measure of central tendency. It can be used with both
discrete and continuous data, although its use is most often
with continuous data
• For example, consider the wages of staff at a factory below:
•The mean salary for these ten staff is $30.7k.

5. Median
• The median is the middle score for a set of data that has
been arranged in order of magnitude. The median is less
affected by outliers and skewed data. In order to calculate
the median, suppose we have the data below:
•Our median mark is the middle mark - in this case, 56

6. Mode
• The mode is the most
frequent score in our data
set. On a histogram it
represents the highest bar in
a bar chart or histogram. You
can, therefore, sometimes
consider the mode as being
the most popular option. An
example of a mode is
presented below:

7. Geometric Mean
• The geometric mean is the average of a set of products, the
calculation of which is commonly used to determine the
performance results of an investment or portfolio.
• It is technically defined as "the nth root product of n numbers.“
• The geometric mean is an important tool for calculating portfolio
performance for many reasons, but one of the most significant is it
takes into account the effects of compounding.
• For volatile numbers, the geometric average provides a far more
accurate measurement of the true return by taking into account
year-over-year compounding that smooths the average.

8. Weighted Mean
•The weighted mean is a type of mean that is calculated by
multiplying the weight (or probability) associated with a
particular event or outcome with its associated quantitative
outcome and then summing all the products together.
• (5* 282.38+4*280.86+3*281.33+2*279.95+5*281.42)/
(5+4+3+2+1)
• 4220.65/15 = 281.38
Days 1 2 3 4 5
Price 282.38 280.86 281.33 279.95 281.42

9. Measurement of Dispersion
•the measure of dispersion shows the scatterings of the data.
It tells the variation of the data from one another and gives a
clear idea about the distribution of the data.
•The measure of dispersion shows the homogeneity or the
heterogeneity of the distribution of the observations.
•A measure of dispersion should be rigidly defined
•It must be easy to calculate and understand
•Not affected much by the fluctuations of observations
•Based on all observations

10. Measurement of Dispersion
Standard
Deviations
Variance

11. Standard deviations
• A standard deviation is the positive square root of the arithmetic
mean of the squares of the deviations of the given values from
their arithmetic mean.
• It is denoted by a Greek letter sigma, σ.
• The square of the standard deviation is the variance. It is also a
measure of dispersion.
• Squaring the deviations overcomes the drawback of ignoring signs
in mean deviations
• Suitable for further mathematical treatment
• Least affected by the fluctuation of the observations
• The standard deviation is zero if all the observations are constant
• Independent of change of origin

12. Variance
• Variance (σ2) is a measurement of the spread between
numbers in a data set.
• It measures how far each number in the set is from the
mean and is calculated by taking the differences between
each number in the set and the mean, squaring the
differences (to make them positive) and dividing the sum of
the squares by the number of values in the set.
• Variance is one of the key parameters in asset allocation.
• The variance of asset returns helps investors to develop
optimal portfolios by optimizing the return-volatility trade-
off in investment portfolios.