Formula and Notation

∑-denotes addition of a set of values X-the variable used to represent the

individual data values N-the number of values in a

population n-the number of values in a sample

Population --complete collection of all elements to be studied

Sample--subcollection of members selected from a population

Mean

Population Sample Weighted

1

N

i

i

x

N

1

i

i

nx

xn

1

1

iWi

i

i

i

kx

xkw

Note

=lower case Greek mu =denotes all values of the population

=pronounced as x- bar =denotes all the data values from a

sample of larger population

x

Med

ian

Popula

tion

Sam

ple

If N is odd

If N is even

If n is odd

If n is even

1( )2

Nx

( 1)2 2

1[ ]2

N Nx x

( 1)2 2

1[ ]2

n nx x x

1( )2

nx x

Sample Problem

Observations: Number of hours spent by the students in studying 2

33362324

2.5

Solving for the Mean

10

1

1 2 3 4. .... 10

1030.5

103.05

xix

nx x x x x

Solving for Median

( 1)2 2

1[ ]21 10 10[ ( 1)]2 2 21( 5 6)21(3 3)23

n nx x x

x x

x x

Mode

3 -- the highest frequency

Further examples

Mean, Median and Mode

Observations=60

87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77, 73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68μ = (87+85+ 79 +….+72+68)/60

= 4751/60

= 79.183

Mean

We now find the median of the population of temperature readings

87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77, 73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68Arrange these 60 measurements in ascending order

65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92

Since N/2 = 30 and both the 30th and 31st values in the list are the same, we obtain median = 78

Median

One further parameter of a population that may give some indication of central tendency of the data is the mode

Define: mode = most frequently occurring value in the population

From the previous data we see:

65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92

That the value 81 occurs 8 times mode = 81

Note! If two different values were to occur most frequently, the distribution would be bimodal. A distribution may be multi-modal.

Mode

Grades of Student 1

Grade

99929091888582

Units

3261433

Solution

1

1

iWi

i

i

i

kx

xkw

99(3) 92(2) 90(6) 91(6) 88(4) 85(3) 82(3)

3 2 6 1 4 3 389.32GPA