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# Empirical rule - Probabilities. Week 5 (1)

## 1. BBA182 Applied Statistics Week 5 (1) Empirical rule - Probabilities

DR SUSANNE HANSEN SARALEMAIL: [email protected]

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DR SUSANNE HANSEN SARAL

1

## 2. Interpretation of summary statistics A random sample of people attended a recent soccer match. The summary statistics (Excel output) about their ages is here below:

People who attended a soccer matchWhat is the sample size?

What is the mean age?

What is the median?

What shape does the distribution of ages

have? (symmetric or non-symmetric)

What is the measure/s for spread in the data?

Is this a large spread?

What is the Coefficient of variation for

this data?

Mean

Median

Mode

Standard Deviation

Sample Variance

Kurtosis

Skewness

Range

Minimum

Maximum

Sum

Count

28.77143

27

23

12.70268

161.358

0.421354

0.854589

53

12

65

1007

35

## 3. Deviations from the normal distribution - Kurtosis

A distribution with positive kurtosis is pointy and a distribution with a negative kurtosis isflatter than a normal distribution

DR SUSANNE HANSEN SARAL, [email protected]

## 4. Positively and negatively skewed

Positive skewed is when the distribution is skewed to the rightNegative skewed is when the distribution is skewed to the left

## 5. Symmetric distribution - Empirical rule

Knowing the mean and the standard deviation of a data set we can extract a lot ofinformation about the location of our data.

The information depends on the shape of the histogram (symmetric, skewed, etc.).

If the histogram is symmetric or bell-shaped, we can use the Empirical rule.

DR SUSANNE HANSEN SARAL, [email protected]

## 6. Probability as Area Under the Curve

The total area under the curve is 1.0, and the curve is symmetric, sohalf (50%) of the data in the data set is above the mean, half (50%) is

below

f(X)

P( X μ) 0.5

0.5

P(μ X ) 0.5

0.5

μ

X

P( X ) 1.0

DR SUSANNE HANSEN SARAL

Ch. 5-6

## 7.

The Empirical RuleIf the data distribution is symmetric/normal, then the interval:

μ 1σ contains about 68% of the values in the population or the

sample

68%

μ

μ 1σ

DR SUSANNE HANSEN SARAL, [email protected]

## 8.

The Empirical Rule(continued)

μ 2σ

contains about 95% of the values in the population or the sample

μ 3σ

contains almost all (about 99.7%) of the values in the population

or the sample

95%

99.7%

μ 2σ

μ 3σ

DR SUSANNE HANSEN SARAL, [email protected]

## 9. Empirical rule: Application

A company produces batteries with a mean lifetime of 1’200 hours and a standarddeviation of 50 hours.

Find the interval for (what values fall into the following interval?):

μ 1σ

μ 2σ

μ 3σ

DR SUSANNE HANSEN SARAL, [email protected]

## 10. Empirical rule: Application

A company produces batteries with a mean lifetime of 1’200 hours and a standarddeviation of 50 hours.

The mean is 1’200 and standard deviation is 50, we find the following intervals:

= 1’200 ± 1x(50) = (1’150 and 1’250)

μ 1σ

= 1’200 ± 2x(50) = (1’100 and 1’300)

μ 2σ

= 1’200 ± 3x(50) = (1’050 and 1’350)

μ 3σ

DR SUSANNE HANSEN SARAL, [email protected]

## 11. Interpretation of the Empirical rule: Lightbulb lifetime example

If the shape of the distribution is normal, then we can conclude :That approximately 68% of the batteries will last between 1’150 and 1’250 hours

That approximately 95% of the batteries will last between 1’100 and 1’300 hours

and

That 99.7% (almost all batteries) will last between 1’050 and 1’350 hours.

It would be very unusual for a battery to loose it’s energy in ex. 600 hours or

1’600 hours. Such values are possible, but not very likely. Their lifetimes would

be considered to be outliers

DR SUSANNE HANSEN SARAL, [email protected]