MBADM813

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Lesson 1: Decision Making Under Uncertainty

Measures of Variability

Now we will discuss measures of variability (variance, standard deviation, and range) using the production line example we used before.

Variance

Variance:: Variance and its related measure, standard deviation, are arguably the most important statistics. Used to measure variability, they also play a vital role in almost all statistical inference procedures.
The formula for variance is given by: sum squared distance from mean, divide by one less than the number of numbers.

Population variance is denoted by...(Lowercase Greek letter “sigma” squared): $σ^{2} = \frac{\sum_{i = 1}^{N} {(x_{i} - μ)}^{2}}{N}$
Sample variance is denoted by...(Lowercase “S” squared): $s^{2} = \frac{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}{n - 1}$

Example: Last Five Weights, Line 1

**Table 1.5. Variance: Time and Weight**
Time (minutes)	Line 1 weight (oz)
544.0	24.85
544.5	25.04
545.0	24.68
545.5	24.83
546.0	24.82

$\bar{X} = \frac{24.85 + 25.04 + 24.68 + 24.83 + 24.82}{5} = 24.84$

$s^{2} = \frac{\sum_{j = 1}^{n} {(X_{j} - \bar{X})}^{2}}{n - 1} = \frac{{(X_{1} - \bar{X})}^{2} + {(X_{2} - \bar{X})}^{2} + \dots + {(X_{n} - \bar{X})}^{2}}{n - 1}$

$s^{2} = \frac{{(24.85 - 24.84)}^{2} + {(25.04 - 24.84)}^{2} + {(24.68 - 24.84)}^{2} + {(24.83 - 24.84)}^{2} + {(24.82 - 24.84)}^{2}}{4}$

Why take the squared difference from the mean?

So that positive and negative differences do not cancel each other out.

Standard Deviation: Square root of the variance

Population standard deviation:

$σ = \sqrt{σ^{2}}$

Sample standard deviation:

$s = \sqrt{s^{2}}$

Example: Last Five Weights, Line 1

**Table 1.6. Standard Deviation Time and Weight**
Time (minutes)	Line 1 weight (oz)
544.0	24.85
544.5	25.04
545.0	24.68
545.5	24.83
546.0	24.82

$s = \sqrt{\frac{\sum_{j = 1}^{n} {(X_{j} - \bar{X})}^{2}}{n - 1}} = \sqrt{\frac{{(X_{1} - \bar{X})}^{2} + {(X_{2} - \bar{X})}^{2} + \dots + {(X_{n} - \bar{X})}^{2}}{n - 1}}$

$s = \sqrt{0.0167397} = 0.12805$

Why take the square root?

Same unit as mean, more meaningful.

One of the simplest measures of spread is the range.

Range: the difference between the two extreme values. It is the measure of spread.
Range = Max - Min

Example: Last Five Weights, Line 1

Find the range for the last five weights from Line 1.

**Table 1.7. Range: Time and Weight**
Time (minutes)	Line 1 weight (oz)
544.0	24.85
544.5	25.04
545.0	24.68
545.5	24.83
546.0	24.82

Range = 25.04 - 24.68 = 0.36

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MBADM813

Main Content

Lesson 1: Decision Making Under Uncertainty

MBADM813:

Lesson 1: Decision Making Under Uncertainty

Measures of Variability

Variance

Example: Last Five Weights, Line 1

Example: Last Five Weights, Line 1

Example: Last Five Weights, Line 1