Variance Math 115b Mathematics for Business Decisions, part II Ekstrom Math 115b

Variance Variance gives a measure of the dispersion of data Larger variance means the data is more spread out Several formulas for different types of variables

Variance: Finite R.V. Finite R.V. Easiest if a table is set up x Given Compute

Variance: Finite R.V. Ex. Find the variance for the following finite R.V., X x fX (x) 1 0.30 3 0.15 5 0.10 7 9

Variance: Finite R.V. First find the mean x fX (x) 1 0.30 3 0.15 5 0.10 7 9

Variance: Finite R.V. Calculate x-value minus the mean x 1 0.30 3 0.15 0.10 7 9 -4 -2 2 4 Calculate x-value minus the mean

Variance: Finite R.V. Calculate the square of x-value minus the mean x 1 0.30 -4 3 0.15 -2 5 0.10 7 2 9 4 16 4 Calculate the square of x-value minus the mean

Variance: Finite R.V. x 1 0.30 -4 16 3 0.15 -2 4 5 0.10 7 2 9 4.80 0.60 0.00 Multiply the squared values by their respective p.d.f. values

Variance: Finite R.V. Variance = 10.8 x 1 0.30 -4 16 4.80 3 0.15 -2 4 0.60 5 0.10 0.00 7 2 9 Variance = 10.8 Add the final column to find the variance

Variance: Finite R.V. Ex. Find the variance for the following finite R.V., X x 1 0.10 3 0.25 5 0.30 7 9

Variance: Finite R.V. First find the mean x fX (x) 1 0.10 3 0.25 5 0.30 7 9

Variance: Finite R.V. Calculate x-value minus the mean x 1 0.10 3 0.25 0.30 7 9 -4 -2 2 4 Calculate x-value minus the mean

Variance: Finite R.V. Calculate the square of x-value minus the mean x 1 0.10 -4 3 0.25 -2 5 0.30 7 2 9 4 16 4 Calculate the square of x-value minus the mean

Variance: Finite R.V. x 1 0.10 -4 16 3 0.25 -2 4 5 0.30 7 2 9 1.60 1.00 0.00 Multiply the squared values by their respective p.d.f. values

Variance: Finite R.V. Variance = 5.2 x 1 0.10 -4 16 1.60 3 0.25 -2 4 1.00 5 0.30 0.00 7 2 9 Variance = 5.2 Add the final column to find the variance

Variance Variance is used to find standard deviation Standard deviation is ALWAYS the square root of variance Standard deviation is represented by sigma or Standard deviation is the “typical amount” of variation from the mean (approx. 2/3 of all data lies within 1 standard deviation of mean)

Variance: Binomial R.V. Shortcut for binomial R.V.

Variance: Binomial R.V. Ex. Suppose X represents to total number of students that pass a particular class in a given semester. If there are 34 students in the class and historically 83% of the students pass, find the standard deviation of X. Soln:

Variance: Binomial R.V. A solution could also be attempted using the BINOMDIST function Recall binomial R.V.’s are finite R.V.’s Complete table as done in previous examples

Variance: Continuous R.V. Similar formula for continuous R.V. Value is found using Integrating.xlsm Recall,

Variance: Continuous R.V. Ex. Suppose is a p.d.f. on the interval [0, 1]. Find the mean of X, the variance of X, and the standard deviation of X. Soln:

Variance: Continuous R.V. Soln:

Variance: Continuous R.V. Ex. Let T be an exponential random variable with parameter . Find and . Soln:

Variance: Continuous R.V. Note that for an exponential random variable, the variance is equal to and the standard deviation is equal to . Ex. Let W be a uniform random variable on the interval [0, 30]. Find and .

Variance: Continuous R.V. Soln.: Estrom Math 115b

Variance: Continuous R.V. Note that for a uniform random variable, the variance is equal to .

Variance: Standardization Different types of variables can have similar parameters (mean & std. dev.) We can transform the variables New variable S defined as

Variance: Standardization We say S is the standardization of X. The mean of S will be 0 The standard deviation of S will be 1.

Variance: Standardization Determining probabilities using standardized values Ex. Suppose X is an exponential random variable with parameter . Determine where S is the standardization of X. Soln: Recall

Variance: Standardization So, Then,

Variance: Sample Formula: A sample is a collection of data from some random variable (finite or continuous)

Variance: Sample Standard deviation of a sample is found by taking the square root of variance Formula: Ex. Find the mean, variance, and standard deviation of the sample 14, 16, 17, 21, 22.

Variance: Sample Soln: Mean: AVERAGE function in Excel

Variance: Sample Variance: VAR function in Excel

Variance: Sample Standard Deviation: STDEV function in Excel

Variance: Sample Why are sample values important? Sometimes unreasonable/impossible to achieve all values for a random variable We can assume and Samples help predict values for the random variable

Variance: Sample Mean Formula: Compares a group of sample means

Variance: Sample Mean Ex. Find the variance and standard deviation of the sample mean for the following data set: 14, 16, 17, 21, 22 Soln:

Variance: Sample Mean How do you interpret ? If there were samples of size 5, the sample mean would be about 18 (from previous calculation) and, on average, the sample mean would be within 1.5166 units about 2/3 of the time.

Variance: Project What to do: Calculate standard deviation of errors (use STDEV function) My standard deviation is about 13.53 We assume mean is 0 even though we calculated a value that was different