QM 250CHAPTER (3) Part 2

Numerical Measures

1st Semester 2021/2021

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Age of QM250 students19 20 21 22 23 19 22 20 21 21 24 26

Grouped data (frequency distribution)

15 up to 20 20 up to 25 25 up to 302 8 1

What is the type of data

Grouped vs. ungrouped data

This chapter can be summarized as the following Type of

data

Ungrouped data

Population

Mean Range

Median Variance

ModeStandard Deviation

Sample

Mean Range

Median Variance

Mode Standard Deviation

Grouped data

Mean Standard Deviation

Measures of Location

Measures of Dispersion

Measures of location for ungrouped data

• Population Mean: The population mean is the sum of all the values in the population divided by the number of values in the population.

• Sample Mean: The sample mean is the sum of all the values in the sample divided by the number of values in that sample.

• MEDIAN: The midpoint of the values after they have been ordered from the minimum to the maximum values.

• MODE The value of the observation that occurs most frequently

Measures of dispersion for ungrouped data• Range: The range is the difference between the maximum and minimum values in a set of

data

• The variance: The variance is the mean of the squared deviations from the arithmetic mean

• Standard deviation: square root of variance

Measures of location for ungrouped data

Example 1

6 10 8 8 5 3

Find: Mean, Median, Model

Example 2

10 10 8 8 5 3

Find: Mean, Median, Model

Example 3

10 9 8 7 5 3

Find: Mean, Median, Model

Example: the weekly overtime hours worked by all the employees at XYZ Saloon are:

1 3 7 12 5 2 3 2 3

• What is the type of data

• Is this is population or sample? Why?

• What is the mean number of overtime hours worked?

• What is the median

• What is the mode

• Calculate σ(𝑥 − 𝜇)

Continue

• Calculate σ(𝑥 − 𝜇)

Continue

Example: Tell me your age:

• What is the type of data

• Is this is population or sample? Why?

• What is the mean, mode and median age of the student?

Outliers or extreme values

If one or two of these values are either extremely large or extremely small compared to the majority of data. the mean might not be an appropriate average to represent the data.

(Mean is affected by extreme values (outliers)

Parameter vs. statistic

• A measurable characteristic of a sample is a statistic

• A measurable characteristic of a population is a

parameter

The following data reflect the score in a subject 20 25 30 22 28 100• Commute mode, median and mode

• What will happen if we deleted the 100 above

Example

A survey has been conducted to check the favorite color of painting your house. Below is the results of 10 people:

White grey white white green blue grey white white

white

What is the mean, mode, median?

Example

Formula

Population

Sample

Advantages (Pros)• Easy to calculate and widely used • It is also known as average,

arithmetic mean or expected value.• The mean is unique• All the data values are used in the

calculation of the mean• The sum of the deviations from the

mean equals zero

Disadvantages (cons)• The mean is influenced by the

extreme value (outliers) • If a distribution is highly skewed,

the mean is probably not a representative measure of central tendency and the median or mode should be used

Two steps to calculate the Median:Step 1: order the dataStep 2: take the Middle

If the number of data is Even: take the average of value with the ranking n/2 and n/2 +1

If the number of data is Odd: take the value with the ranking (n+1)/2

• The median is, therefore, unaffected by extreme values• The median is the value in the middle of a set of ordered data• 50% of the observations are larger than the median and 50% are lower• It is unique. There is only one Median• At least the ordinal scale of measurement is required

• The Mode is unaffected by extreme values• It can be used for qualitative data • There might be one Mode, more than one or no Mode (This

is disadvantage)

Relative Positions of Mean, Median, and Mode