Measures of Central Tendency (Raw Dats)

1. MEASURES OF CENTRAL
TENDENCY
PRESENTED BY: ALONA HALL

2. OBJECTIVES
From this lesson, we hope to:
 Define the concept related to measures of central
tendency
 Demonstrate how each measure of central
tendency (mean, mode and median) is calculated
from raw data

3. THE CONCEPT OF MEASURES OF CENTRAL TENDENCY
• The measures of central tendency are known as the mean, the
mode and the median. They are all known as statistical averages
but the arithmetic mean is the most popular average.
• Each measure, when determined, gives the statistician/the
observer information about what conclusions can be made about
the set of data which it represents.

4. FOR EXAMPLE, IF A LARGE POPULATION OR A LARGE SAMPLE FROM A
POPULATION IS BEING OBSERVED, ONE OF THE THREE SITUATIONS
BELOW MAY OCCUR AND THERE ARE OTHERS AS WELL
Depending on the situation there are particular interpretations and conclusions that can be made.
NB: These diagrams do not apply when the number of observations being observed is small

5. • In other words, the diagrams shown in the previous slide would
not represent raw data since a large sample or population
would need to be organized in order to determine such
measures.
Bear in mind that raw data is data that has not been organized.

6. • Find the mean, mode and median of the
following heights (in centimetres):
a)167,162, 154, 155, 182, 191
b)159,155, 154, 152,156,157,152
EXAMPLE

7. THE MEAN
The mean (or x bar) is calculated using the following formula:
𝑥 =
𝑥
𝑓
=
𝑥
𝑛
(for raw data)
Where 𝑥 = 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛
𝑥 = 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
𝑓 = 𝑛 𝑚𝑒𝑎𝑛𝑠 ′𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 𝑜𝑟 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠′

8. EXAMPLE (CONT’D)
Thus, the mean of: 167,162, 154, 155, 182, 191
is calculated by substituting into the formula 𝑥 =
𝑥
𝑓
=
𝑥
𝑛
i.e. 𝑥 =
167+162+154+155+182+191
6
=
1011
6
= 168.5

9. SIMILARLY
The mean of: 159,155, 154, 152,156,157,152 is 155. Did
you get it?
First, write down the formula: 𝑥 =
𝑥
𝑓
=
𝑥
𝑛
Then, substitute: 𝑥 =
159+155+154+152+156+157+152
7
= 155

10. NOW, LET’S GO TO THE MODE
The mode is the observation that appears most frequently.
Thus, for a) 167,162, 154, 155, 182, 191
There is NO mode.
But for b) 159,155, 154, 152,156,157,152
The mode is 152 since 152 appears twice which is more
frequently than the other observations which appear only
once.

11. FINALLY, LET’S LOOK AT THE MEDIAN (DENOTED 𝑄2)
This is the observation that appears in the centre after having arranged
the values in ascending or descending order.
The way to determine the median is different depending on whether
there is an even number of observations or an odd number of
observations, for example:
a) 167,162, 154, 155, 182, 191 (There are 6 heights. 6 is an even
number)
b) 159,155, 154, 152,156,157,152 (There are 7 heights. 7 is an odd
number)

12. LET’S DETERMINE THE MEDIAN
a) 167,162, 154, 155, 182, 191
 First: arrange in ascending or descending order:
If the number of observations is even then two values will be in the
centre
154, 155, 162, 167, 182, 191
In this case, find the average of those two ‘middle observations’
Thus, 𝑄2 =
162+167
2
= 164.5

13. However, if the number of observations is odd, then only one
observation will be the central value. That observation will
be taken as the median.
b) 159,155, 154, 152,156,157,152
In descending order, we get:
159, 157, 156, 155, 154, 152, 152.
Thus, 𝑄2 = 155