1. Statistical analysis and its application
The Regional Workshop on the Practical Application of Payment Systems Operations and Oversight Windhoek – October 2017
Statistical analysis and its application
Hopolang Phillip Mashele

2. Outline Basic statistics: Concepts Basic statistics: Excel formulas
Basic Probability Theory
Normal Distribution
Exercises

3. Basic statistics: Concepts
Random Variables:
When the numerical value of a variable is determined by a chance event, that variable is called a random variable.
Random variables can be discrete or continuous:
Discrete random variables:
Discrete random variables take on integer values, usually the result of counting. Suppose, for example, that we flip a coin and count the number of heads.
The number of heads results from a random process - flipping a coin.
And the number of heads is represented by an integer value - a number between 0 and plus infinity.
Therefore, the number of heads is a discrete random variable. 3

4. Basic statistics: Concepts (Cont…)
Continuous random variables :
Continuous random variables, in contrast, can take on any value within a range of values.
For example, suppose we flip a coin many times and compute the average number of heads per flip.
The average number of heads per flip results from a random process - flipping a coin. And the average number of heads per flip can take on any value between 0 and 1, even a non-integer value.
Therefore, the average number of heads per flip is a continuous random variable. 4

5. Basic statistics: Concepts (Cont…)
Mean: the expectation of a random variable X having possible values
is defined by or
Variance and Standard Deviation: the variance or the standard deviation provides a measure of extent of the dispersion in the values of the random variable around the mean:
Standard Deviation =
Covariance: covariance measures the extent of dependence of variables X and Y:
Note that 5

6. Basic statistics: Concepts (Cont…)
Correlation Coefficient: correlation coefficient also measures the extent of dependence of variables X and Y:
Note that and
Median: it is the midpoint of a data set if the data is arranged in ascending or descending order.
Mode: it is the value that occurs most frequently in a data set. 6

7. Basic statistics: Excel formulas
MEAN
MEDIAN:
CORRELATION:
COVARIANCE:
MODE:
MAXIMUM:
MINIMUM:
STANDARD DEVIATION: 7

8. Examples: Basic statistics
The following are the forecasted payoffs for the shares of two companies:
Calculate the expected returns of both companies.
Calculate the standard deviations of both companies.
Calculate the covariance between these companies.
Calculate the correlation coefficient between these companies.
State of Economy
Probability if State occurs
Anglo Gold (mining)
BMW (luxury cars)
Recession
30%
150%
50%
Normal
40%
100%
Boom 8

9. Examples: Basic statistics (cont…)
Answers:
Expected return:
E(Anglo) = (0.3)(150) + (0.4)(100) + (0.3)(50) = 100%
E(BMW) = (0.3)(50) + (0.4)(100) + (0.3)(150) = 100%
Variance:
Standard deviation: 9

10. Examples: Basic statistics (cont…)
Covariance:
cov(BMW, Anglo) = (0.3)( )( ) + (0.4)( )( ) + (0.3)( )( ) = -1.5
The negative covariance indicates that the prices of BMW and Anglo Gold move in opposite directions.
Correlation coefficient:
This suggests that the two stocks are perfectly negatively correlated with each other. 10

11. Examples: Basic statistics (cont…)
Calculate the median and the mode of the following data set:
Answers:
Mode = 28
To calculate the median, we may arrange the data set in ascending order:
Median = ½[23 +25] = 24
Open Advanced_Excel.xls 11

12. Basic Probability Theory
Probability is defined as the chance that a particular event will occur ,i.e.,
where X = the number of outcomes for the event, and T = the total number of possible outcomes.
Hence the probability of picking a red from a deck of cards is 26/52 since there are 26 red cards in a deck of 52 cards.
Mutually exclusive means that the two events cannot both occur at same time.
For example, events ‘Head’ and ‘Tail’ of a coin are mutually exclusive.
refers to the event “A and B”.
refers to the event “either A or B”. 12

13. Some important theorems of probability
For any event A,
The impossible event has a probability of zero
If are mutually exclusive events and , then
If A and B are any two events (not mutually exclusive), then 13

14. Examples: Basic Probability Theory
A card is drawn from a deck of cards. What is the probability that it is a 5 or a heart (H)?
Answer:
Since drawing a 5 or drawing a heart are not mutually exclusive events, applying Theorem 4, we obtain: 14

15. Examples: Basic Probability Theory15

16. Examples: Basic Probability Theory
A ball is drawn at random from an urn containing 4 red (R), 4 white (W) and 4 green (G) balls. What is the probability that the ball drawn is
White (W)?
Red (R) or White (W)?
Answers:
Since we cannot have R and W at the same time
Therefore, applying Theorem 4,
Or apply Theorem 3. 16

17. Probability Distributions
A probability distribution is a table or an equation that links each possible value that a random variable can assume with its probability of occurrence.
Discrete Probability Distributions: The probability distribution of a discrete random variable can always be represented by a table. For example, suppose you flip a coin two times. This simple exercise can have four possible outcomes: HH, HT, TH, and TT. Now, let the variable X represent the number of heads that result from the coin flips. The variable X can take on the values 0, 1, or 2; and X is a discrete random variable.
For a discrete distribution, if x cannot occur, or
if it can occur. 17

18. Probability Distributions (cont…)
For example, the probability of it raining on 34 days in June is zero because there are only 30 days in June. However, the probability of it raining 25 days in June has some positive value.
The table below shows the probabilities associated with each possible value of the X. The probability of getting 0 heads is 0.25; 1 head, 0.50; and 2 heads, Thus, the table is an example of a probability distribution for a discrete random variable.
Given a probability distribution, you can find cumulative distribution probabilities.
For example, the probability of getting 1 or fewer heads [ P(X < 1) ]
is F(x) = P(X = 0) + P(X = 1), which is equal to = 0.75. 18

19. Probability Distributions (cont…)
Continuous Probability Distributions: The probability distribution of a continuous random variable is represented by an equation, called the probability density function (pdf). All probability density functions satisfy the following conditions:
The random variable Y is a function of X; that is, y = f(x).
The value of y is greater than or equal to zero for all values of x.
The total area under the curve of the function is equal to one.
For a continuous distribution, even though x can occur.
For example, the probability of receiving exactly two inches of rain in June is zero because two inches is a single point in an infinite range of possible values. Conversely, the probability of the amount of rain being between and inches has some positive value. 19

20. Probability Distributions (cont…)
The charts below show two continuous probability distributions. The chart on the left shows a probability density function described by the equation y = 1 over the range of 0 to 1 and y = 0 elsewhere. The chart on the right shows a probability density function described by the equation y = x over the range of 0 to 2 and y = 0 elsewhere. The area under the curve is equal to 1 for both charts. 20

21. Probability Distributions (cont…)
The probability that a continuous random variable falls in the interval between a and b is equal to the area under the pdf curve between a and b, i.e.,
where and
Note that
For example, in the first chart above, the shaded area shows the probability that the random variable X will fall between 0.6 and That probability is And in the second chart, the shaded area shows the probability of falling between 1.0 and 2.0. That probability is 0.25. 21

22. Normal Distribution
Normal Distribution: in the area of investment and portfolio management, the normal distribution plays a central role in portfolio theory.
Normal Distribution has the following key properties:
The random variable X is normally distributed with mean and variance , i.e.,
It is symmetric about the its mean: Skewness = 0.
Kurtosis = 3.
Approximately 50% of all observations fall in the interval % in the interval % in the interval and 99.5% in the interval
Probability Density Function: 22

23. Normal Distribution (cont…)
Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set (dark blue), while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99%. 23

24. Normal Distribution (cont…)
Skewness: it refers to the extent to which a distribution is not symmetrical.
Excel formula:
The mean is pulled in the direction of the skew:
Symmetrical distribution: Mean = Median = Mode and Skewness = 0
Positive/right skew: Mean > Median > Mode
Negative/left skew: Mean < Median < Mode
Kurtosis: it is measure of the degree to which a distribution is more or less peaked than a normal distribution.
Leptokurtic describes a distribution that is more peaked than a normal distribution
Platykurtic refers to a distribution that is less peaked or flatter than a normal distribution 24

25. Normal Distribution (cont…)
Standardization is the process of converting an observed value x for a random variable X to its z test statistic:
Standard normal distribution is a normal distribution that has been standardized so that
Excel formulas: 25

26. Normal Distribution (cont…)26

27. Normal Distribution (cont…)
Example 1:
Assume that the annual Earnings Per Share (EPS) of a large sample Central Banks in Africa are normally distributed with the mean of $6.00 and the standard deviation of $2.
What is the probability that an observed EPS will fall between $2.00 and $8.00?
What is the probability that the EPS values are at least $9.70?
Answers:
Using 27

28. Normal Distribution (cont…)
Using 28

29. Normal Distribution (cont…)
Example 2:
The life of a fully-charged cell phone battery is normally distributed with a mean of 14 hours and a standard deviation of 1 hour.
What is the probability that a battery lasts at most 13 hours?
Answer:
Using
Open normal.xls 29

30. THANK YOU! Questions? 30