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Probabilistic and Statistical Techniques

Lecture 4

Dr. Nader Okasha

Descriptive measures of data:

1. Measure of Center

2. Measure of Variation

3. Measure of Position

Measure of Center

The value at the center or middle of a data set.

The purpose of a measure of center is to pinpoint the center of a set of values.

Arithmetic Mean (Mean)

The measure of center obtained by adding the values and dividing the total by the number of values

The mean is affected by outliers

Notation

denotes the sum of a set of values.

x is the variable usually used to represent the individual data values.

n represents the number of values in a sample.

N represents the number of values in a population.

Population Mean

The population mean is the sum of all values in the population divided by the number of the values in the population.

is pronounced ‘mu’ and denotes the mean of all values in a population

N

x

Example

There are 12 automobile manufacturing companies in the United States. Listed below is the no. of patents granted by U.S. government to each company in a recent year

No. of Patents Company No. of Patents Company210 Mazda 511 General Motors

97 Chrysler 385 Nissan

50 Porsche 275 DaimlerChrysler

36 Mitsubishi 257 Toyota

23 Volvo 249 Honda

13 BMW 234 Ford

Example

This is an example of a population mean because we are considering all the automobiles manufacturing companies obtaining patents.

=195

Sample Mean

The sample mean is the sum of all the sampled values divided by the total number of sampled values.

is pronounced ‘x-bar’ and denotes the mean of a set of sample values

n

xx

x

Sample Mean Example

SunCom is studying the number of minutes used monthly by clients in a particular cell phone rate plan. A random sample of 12 clients showed the following number of minutes used last month

90 77 94 89 119 112 91 110 100 92 113 83

What is the arithmetic mean number of minutes used?

Sample mean = 97.5 minutesx

Median

The middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

It is not affected by an extreme value

If the number of values is even, the median is found by computing the mean of the two middle numbers.

If the number of values is odd, the median is the number located in the exact middle of the list.

5.40 1.10 0.42 0.73 0.48 1.10

0.42 0.48 0.73 1.10 1.10 5.40

(in order - even number of values)

MEDIAN is = 0.9150.73 + 1.102

Example

Mode

The value that occurs most frequently Mode is not always unique A data set may be:

Bimodal

Multimodal

No Mode

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Example

a. 5.40 1.10 0.42 0.73 0.48 1.10

b. 27 27 27 55 55 55 88 88 99

c. 1 2 3 6 7 8 9 10

Mode is 1.10

Bimodal - 27 & 55

No Mode

Midrange

The value midway between the maximum and minimum values in the original data set

Midrange = maximum value + minimum value

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Example

Table 3.1Table 3.2

Data Set IData Set II

For each data set determine: Mean Median Mode

Example

Solution

Mean from a Histogram

Use interval midpoint of classes for variable x

Interval midpoint Number of counts

Example

Interval limits

Interval Mid point x

No. of counts f f . x

21 - 30 25.5 28 71431 - 40 35.5 30 106541 - 50 45.5 12 54651 - 60 55.5 2 11161 - 70 65.5 2 13171 - 80 75.5 2 151

Sum 76 2718

76.3576

2718.

f

xfx