The mean values

Part 1 THE MEAN VALUES

СHAPTER QUESTIONS

Measures of location
Types of means
Measures of location for ungrouped data
-

Properties to describe numerical data:
Central tendency
Dispersion
Shape

Measures calculated for:
Sample data
Statistics
Entire population
Parameters

Measures of

Measures of location include:
Arithmetic mean
Harmonic mean
Geometric mean
Median

Grouped and Ungrouped

UNGROUPED or raw data refers to data as

What is the mean?

The mean - is a general indicator characterizing

Statistics derive the formula of the means of the formula of

There are the following types of mean:
If z = -1

The higher the degree of z, the greater the value of

There are two ways of calculating mean:
for ungrouped data -

Types of means

Arithmetic mean

Arithmetic mean value is called the mean value of

Characteristics of the arithmetic mean

The arithmetic mean has a number

2. If the data values (Xi) divided or multiplied by a

3. If the frequency divided by a constant number, the mean

4. Multiplying the mean for the amount of frequency equal to

5.The sum of the deviations of the number in a

Measures of location for ungrouped data

In calculating summary values for a

Measures of location for ungrouped data

ARITHMETIC MEAN
-

sum of observations
number of observations

Population mean =

Measures of location for

Example - The sales of the six largest restaurant chains are

MEDIAN for ungrouped data

The median of a data is the middle

MEDIAN
Every ordinal-level, interval-level and ratio-level data set has a

Position of median

If n is odd:
Median item number

Example

The median number of people treated daily at the emergency

MODE for ungrouped data
Is the observation in the data set that

The simple mean of the sample of nine measurements is given

−4 −3 2 2 5 5 5 6 8

Median item number =
(n+1)/2 = (9+1)/2 = 5th measurement

1

2

3

4

5

6

7

8

9

Median =

Determine the median of the sample of ten measurements.
Order the measurements

Example

Determine the mode of the sample of nine measurements.
Order the measurements

Determine the mode of the sample of ten measurements.
Order the measurements

Is used if М = const:
Harmonic mean is also called the

Geometric mean for ungrouped data

This value is used as the average

Where П – the multiplication of the data value (Xi).
n

For example, the known data about the rate of growth of

Measures of location for grouped data

ARITHMETIC MEAN
Data is given

Example

There are data on seniority hundred employees in the table

Average seniority employee is:

Harmonic mean for grouped data

Harmonic mean - is the reciprocal of

Harmonic mean for grouped data

Harmonic mean is calculated by the formula:
where

Example

There are data on hárvesting the apples by three teams

is calculated by the formula:
Where fi – frequency of the data

Calculate the geometric mean. It is 127,5% percent:

Geometric mean for grouped

Measures of location for grouped data

MEDIAN
Data is given in

Measures of location for grouped data

MODE
Class interval that has the

Mode is calculated by the formula:
where хМо – lower boundary of

To calculate the mean for the sample of the 48 hours:
Determine

Number of Number of xi
calls hours fi
[2–under 5) 3 3,5

To calculate the for the sample median of the 48: hours:
determine

Number of Number of
calls hours fi F
[2–under 5)

Measures of location for grouped data

Example – The following data

To calculate the for the sample mode of the 48 hours
The

We substitute the data into the formula:
Mo = 12,3
So, the most

Measures of location for grouped data

Example – The following data

Relationship between mean, median, and mode

If a distribution is symmetrical:
the mean,

EXAMPLE

Consider a study of the hourly wage rates in three different

So we have three 100-element samples, which have the same average

The histogram for company I (left chart) is symmetric. The histogram

Knowing the median, modal and average values enables us to resolve

We obtain the following relevant indicators (measures) of asymmetry:
Index of

Example

The weighted arithmetic mean