measures of location it is often useful to obtain summaries of certain aspects of the data. most...

Measures of location

It is often useful to obtain summaries of certain aspects of the data.

Most simple summary measurements can be divided into two types;

firstly quantities which are “typical” of the data (with respect to frequency),

and secondly, quantities which summaries the variability of the data.

The former are known as measures of location and thelatter as measures of spread.

Suppose we have a sample of size n of quantitative data. We will denote the measurements by x1;x2; : : :xn.

Central tendency is the middle point of a distribution. Measures of the central tendency is called as Measure of location.

BIC Prepaid By:Rajyagor Bhargav

This is the most important and widely used measure of location. The sample

mean of a set of data is=(x1+x2+….+xn)/ n

=1/n∑n

i=1xi

This is the location measure often used when talking about the average of a set of observations.

Mean (Arithmetic mean)


e.g.Generator (for power supply)

1 2 3 4 5 6 7 8 9 10

Day out of service 7 23 4 8 2 12 6 13 9 4

Arithmetic mean= 7+23+4+8+2+12+6+13+9+4/10=88/10=8.8 days

In this 10-months period, the generators were out of service for an average of 8.8 days.


Conventional symbols A sample of population consist of n

observations with a mean of The notation is different when we are

computing measures for entire population.

The mean of a population is symbolized by µ.

The number of elements in population is denoted by N.


Population arithmetic mean

µ = ∑x / N Where ∑x = sum of all observation N = number of elements in the

population.


Sample arithmetic mean

= ∑x / n Where ∑x = sum of all observation n = number of elements in the sample.


Calculating the mean from ungrouped data.STUDENTS 1 2 3 4 5 6 7

Score

9 7 7 6 4 4 2

= ∑x / n= 9+7+7+6+4+4+2 /7=39/7=5.6 points per students (sample mean)


Calculating mean from Grouped Data

Data are grouped by class. Each value of an observation falls

some ware in one of the class.


Sample arithmetic mean for grouped data.

= ∑(f X x) / n Where

= sample mean ∑= symbol meaning “the sum of” f = frequency (number of observations) in

each class x = midpoint for each class in the sample. n = number of total observation in the

sample.


We first calculate the midpoint of each class.

Midpoint = Rounding of (starting of class + ending of class )/2

∑(f X x) = multiply each midpoint by the frequency of observations in that class, sum all results, and

Divide the sum by total number of observation in sample.


e.g.

Average Monthly salary of 600 customers of a particular bank.

Class (Dollars) Frequency

0 – 49.99 78

50.00 – 99.99 123

100.00 – 149.99 187

150.00 – 199.99 82

200.00 – 249.99 51

250.00 – 299.99 47

300.00 – 349.00 13

350.00 – 399.99 9

400.00 – 449.00 6

450.00 – 499.00 4

Total customers 600


Answer


To simplify our calculation of a mean from grouped data using a technique called coding.

In this method we eliminate the problem of large or difficult midpoints.

Instead of using actual midpoint to perform our calculation,

we assign small value consecutive whole numbers called codes to each of the midpoints.

Coding


The integer zero can be assigned anywhere, but to keep integer small we will assign zero to midpoint in the middle of the frequency distribution.

Then we can assign negative integer to values smaller then midpoint and positive integers to those larger.


e.g.

Class Midpoint Code

0 – 7 3.5 -2

8 – 15 11.5 -1

16 – 23 19.5 <- x0 0

24 – 31 27.5 1

32 – 39 35.5 2

40 – 47 43.5 3


Sample arithmetic mean of grouped data using code.

= x0 + w (∑(u X f) / n) Where

= mean of sample x0 = value of the midpoint assigned the code 0 w =numerical width of the lass interval. u = code assign to each class. f = frequency or number of observation in each

class. n = total number observation in the sample.


Answer


Advantages

It represent a single number for whole data set.

Its concept is familiar to all people and clear to all.

every data set has a unique mean and easy to calculate.

Mean is useful for performing statistical procedure like co0mparing mean from several data set.


Disadvantages

Reflect all the values in dataset. Although the mean is reliable in that it

reflects all the values in the dataset, it may also affect by extreme value that are not representative of the rest of observation.

MEMBER 1 2 3 4 5 6 7TIMES IN MINUTE 4.2 4.3 4.7 4.8 5.0 5.1 9.0


Calculate the mean for above data= 4.2+4.3+4.7+4.8+5.0+5.1+9.0 / 7= 37.1/7=5.3 minutes ----- population meanIf we compute mean for 6 members, and

exclude 7th . The answer for mean is 4.7.It would more appropriate to calculate

mean without including such a extreme values.


Use every data point in our calculation. We encounter with each and every

data of our data set. So, it is more difficult with vast

amount of data set.


Unable to compute for open ended class. We are unable to compute the

mean for a data set that has open-end class like >100 or <0


Weighted mean

The weighted mean enables us to calculate an average that takes in to account the importance of each value to the overall total.

w = ∑(w X x) / ∑w

where = symbol for the weighted meanw = weight assign to each observation.

∑(w X x) = sum of the weight of each element times that with element ∑w = sum of all weight


Example

LABUR INPUT IN MANUFACTURING COMPANY

LABOUR HOURS PER UNIT OF OUTPUT

Grade of Labor

Hourly Wages(x)

Product1 Product2

Unskilled $5.00 1 4

Semiskilled $7.00 2 3

Skilled $9.00 5 3


A sample arithmetic mean of the labor rate.

= ∑x / n = $5 + $7 +$9 /3 = $21/3 = $7.00 ---per hour

Using this average rate, we would compute the labor cost of one unit of product 1 to be $7(1+2+5) = $56 and same for product2 will be $70 which are incorrect.


To solve this problem we have to consider weight for each observation.

For product1= ($5 X 1) + ($7 X 2) + ($9 X 5)= $64 / 8= $8 --per hour

For product2= ($5 X 4) + ($7 X 3) + ($9 X 3)= $68 / 10

= $6.8 ---per hour


W = ∑(w x x)

= (1/8 x $5) + (2/8 x $7) + (5/8 x $9)1/8 + 2/8 + 8/8

= $8 / 1

= $8.00 / hour


do exercises A mini store advertises ,”if our average price are not equal

or lower than everyone else's, you get it free.” one of the customer came into the store one day and threw on the counter bills of sale of six items she bought from competitor for an average price less then that store. The item cost

$1.29 $2.97 $3.49 $5.00 $7.50 $10.95Store’s price for the same six are

$1.35 $2.89 $3.19 $4.98 $7.59 $11.50Store’s owner told the customer, “my ad refers to a

weighted average price of these items. our average is lower because our sales of these items have been:” 7 9 12 8 6 3

Is store’s owner getting himself into or out of trouble by talking about weighted average?


Geometric mean

some times we dealing with quantities that changes over a period of timewe need to know an average rate of change.Such as an average growth rate over a period of several year.In such cases, the simple arithmetic mean is inappropriate.we need to find the geometric mean simply called the G.M.


How to calculate G.M.

G.M. = n product of all x values

E.g.

Growth of $100 deposit in a saving account

Year Interest Rate

Growth Factor

Saving at End of year

1 $7 $1.07 $107.00

2 8 1.08 115.56

3 10 1.10 127.12

4 12 1.12 142.37

5 18 1.18 168.00


Geometric mean

5G.M. = 1.07 x 1.08 x 1.10 x 1.12 x 1.18

= 5 1.679965

= 1.1093 Average growth factor


measures of location it is often useful to obtain summaries of certain aspects of the data. most...

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