The document discusses various measures of central tendency including the arithmetic mean, median, mode, geometric mean, and harmonic mean. It provides formulas and examples for calculating the arithmetic mean and median. Specifically, it defines the arithmetic mean as the sum of all values divided by the total number of values. An example is shown of calculating the mean for a set of test scores by placing them in a frequency distribution table and using the formula Σfx/Σf. The median is defined as the middle value when values are arranged in order, or the average of the two middle values for an even distribution. A formula is given for finding the median of grouped data using cumulative frequencies.
2. Arithmetic Mean
O Definition: The Arithmetic mean also
popularly referred to as the ‘mean’ is
the average series of figures or values.
O It is obtained by dividing the sum of
these figures by the total number
figures or values.
O It is also the average of a collection of
observation.
O It is the most popularly used measure
of central tendency.
3. Formula for calculating arithmetic
meanOArithmetic mean 𝑥 =
Σ𝑥
𝑛
where
OX = arithmetic mean
O∑ =represents a Greek letter denoting ‘sum of’
OX = a series of figures in a given data
On = number of figures or elements
4. Mean of grouped data
O The arithmetic mean can also be
prepared for grouped data. In this
case, the class mark (mid points) of
the individual class interval can be
used for the x-column.
O Formula: Arithmetic mean 𝑥 =
Σ𝑓𝑥
Σ𝑓
5. Example 1
O Calculate the mean of the following marks
scored by students in an economics
examination
8, 31, 45, 38, 22, 28, 16, 51, 65, 48, 6, 24, 18,
12, 16, 48, 38, 50, 44, 6, 18, 16, 24, 32, 36, 26,
14, 20, 12, 18
O Solution
O Identify the numbers that occur in the set and
arrange the number in a frequency distribution
table
O Arrange the number starting from the smallest
to highest number
6. Continued from last slide
OUse a class interval of 0 – 9, 10 – 19, 20 – 29,
etc.
OPrepare a frequency table.
Scores
(Grouping
s)
x
Class
Mark
Tally or
Events
f
Frequenc
y
(fx)
0 – 9 4.5 ||| 3 13.5
10 – 19 14.5 |||| |||| 9 130.5
20 – 29 24.5 |||| | 6 147.0
30 – 39 34.5 |||| 5 172.5
40 – 49 44.5 |||| 4 178.0
50 – 59 54.5 || 2 109.0
60 – 69 64.5 | 1 64.5
⅀f = 30 ⅀fx =815
8. The median
O The median is defined as an average
which is the middle value when figures are
arranged in order of magnitude.
O In an even distribution, the median is the
average of two middle numbers i.e. the
median of a distribution is the middle
value when the observations are arranged
in other of magnitude starting with either
the smallest or the largest number.
O The median is therefore the value of the
middle item.
9. Median of grouped data
O When a grouped data is involved,
cumulative frequency is used.
O This is used when items or values are
large and arranging them in ascending
order may not work. The formula would
now be
O Median = 𝑁+1
2
th member for odd number
of items where N is odd.
O Median =
𝑁
2
𝑡ℎ+
𝑁
2
+1
2
member for even
number of items i.e. where N is even.
