Central Tendency

2.  Introduction:
In statistics, a central tendency is a central or typical value for a probability distribution.
Simpson and Kafka , ‘a measure of central tendency is typical
value around which other figures aggregate.’
Croxton and Cowden “An average is a single value within the
range of the data that is used to represent all the values in the series. Since an
average is somewhere within the range of data, it is sometimes called a measure of
central value.”

3.  What is central tendency?
• Central tendency is defined as “the statistical measure that identifies a single value as representative of an entire
distribution.”
• It aims to provide an accurate description of the entire data. It is the single value that is most
typical/representative of the collected data.
• Central tendency have divided in Four categories like-
1. Mean
2. Median
3. Mode
4. Midrange

4. 1. What is Mean?
• Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total
number of numbers.
• Mean = (Sum of all the observations/Total number of observations)
• There are three methods to find the mean :
(i) Direct method: In individual series of observations x1, x2,… xn the arithmetic mean is obtained by following
formula.
Mean=
𝑥1
+𝑥2
+𝑥3
+𝑥4
........+𝑥𝑛
𝑛
(ii)Short-cut method: This method is used to make the calculations simpler.
Let A be any assumed mean (or any assumed number), d the deviation of the
arithmetic mean, then we have
Mean= A+
𝑓𝑑
𝑁
{d=(x-A)}

5. (iii)Step deviation method: If in a frequency table the class intervals have equal width, say i than it is convenient to
use the following formula.
M=A+
𝑓𝑢
𝑛
x i
where u=(x-A)/i ,and i is length of the interval, A is the assumed mean.
Example: Class= 20-30, 30-40, 40-50, 50-60, 60-70
Frequency= 8 , 26 , 30 , 20 , 16
Solution:
Class Mid
Value(x)
f fx d=(x-A)
A=45
fd
20-30 25 8 200 -20 -160
30-40 35 26 910 -10 -260
40-50 45 30 1350 0 0
50-60 55 20 1100 10 200
60-70 65 16 1040 20 320
Total N= 100 𝑓𝑥= 4600 𝑓𝑑=100
u=(x-A)/i
i=10
fu
-2 -16
-1 -26
0 0
1 20
2 32
𝑓𝑢=10

6. • By Direct method- If we have to find the mean of any
M=( 𝑓𝑥 )/N dataset then,
= 4600/100= 46. Suppose the dataset is,
• By Short cut method- 2,4,6,8,10,12
Let Assumed mean A=45. So,the mean value
M= A+( 𝑓𝑑)/N =𝑥 =
2+4+6+8+10+12
6
=7.
= 45+(100/100)= 46.
• By Step deviation method-
Let the Assumed mean A= 45
M= A+ i( 𝑓𝑢)/N
=45+ 10 x (10/100)
=46

7. • Types of Mean:
• Arithmetic Mean: When you add up all the values and divide by the number of values it is called Arithmetic Mean.
Ex- 3, 5, 9, 5, 7, 2 ; the arithmatic mean value=
3+5+9+5+7+2
6
=5.16.
• Geometric Mean: It can be found by multiplying all the numbers in dataset and take the nth root for the obtained
result.
Ex- 2,4,6,8,10 ; the geometric mean value=
5
2 𝑋 4𝑋 6 𝑋 8 𝑋 10 = 5.2103
• Harmonic Mean: The Harmonic Mean (HM) is defined as the reciprocal of the average of the reciprocals of the data
values. It is based on all the observations, and it is rigidly defined.
Ex- 3,5,7,9,11 ; the hermonic mean value= n / [(1/x1)+(1/x2)+(1/x3)+(1/x4)+(1/x5)]
= 5/[(1/3)+(1/5)+(1/7)+(1/9)+(1/11)]
= 5.693
Besides, The relationship Between Arithmetic Mean(A), Geometric Mean(G) and Harmonic Mean(H) is-
G= 𝑨𝑯 or, G2 = AH
• Weighted mean: A weighted mean is a kind of average. Instead of each data point contributing equally to the final
mean, some data points contribute more “weight” than others. If all the weights are equal, then the weighted mean
equals the arithmetic mean (the regular “average” you’re used to). Weighted means are very common in statistics,
especially when studying populations.
To find the weighted mean:*Multiply the numbers in your data set by the weights. *Add the results up.

8. • Formula of weighted mean:
Weighted mean = Σwx/Σw where, w= the weights,𝑥= the mean value of data
Example: Find weighted mean for following data set x= quantity
w = {2, 5, 6, 8, 9}, x = {4, 3, 7, 5, 6}
Given data sets w = {2, 5, 6, 8, 9}, x = {4, 3, 7, 5, 6} and N = 5
According to the formula,
Weighted mean = ∑(weights × quantities) / ∑(weights)
= (w1x1 + w2x2 + w3x3 + w4x4 + w5x5) / (w1 + w2 + w3 + w4 + w5)
= (2 × 4 + 5 × 3 + 6 × 7 + 8 × 5 + 9 × 6) / (2 + 5 + 6 + 8 + 9)
= ( 8 + 15 + 42 + 40 + 54) / 30
= 159 / 30
= 5.3
Therefore, the weighted mean is 5.3.
• Truncated mean: A truncated mean (sometimes called a trimmed mean) is similar to a “regular” mean (average),
but it trims any outliers. Outliers can affect the mean (especially if there are just one or two very large values), so
a trimmed mean can often be a better fit for data sets with erratic high or low values or for extremely skewed
distributions.
Example: Find the trimmed 20% mean for the following test scores: 60, 81, 83, 91, 99.
Step 1: Trim the top and bottom 20% from the data. That leaves us with the middle three values:
60, 81, 83, 91, 99.
Step 2: Find the mean with the remaining values. The mean is (81 + 83 + 91) / 3 ) = 85

9. • Inter-quartile mean: The interquartile mean (IQM) is the mean of the middle 50 percent of data in a data set.
Unlike the “regular” arithmetic mean, it is resistant to outliers.
Example: Find the IQM for the following data set:
5 ,6,17, 30, 44, 55, 56, 8, 9, 11, 13, 15, 1 3, 16, 65
Step 1: Sort the data from smallest to largest:
1, 3, 5, 6, 8, 9, 11, 13, 15, 16, 17, 30, 44, 55, 56, 65
Step 2: Discard the bottom 25% and top 25% of numbers. In other words, split the data set into quarters and
remove the top and bottom quarters:
1 3 5 6 | 8 9 11 13 | 15 16 17 30 | 44 55 56 65
Step 3: Find the mean of the remaining numbers:
(8 + 9 + 11 + 13 + 15 + 16 + 17 + 30 ) / 8 = 14.875
• Tri mean: A trimean is a number that represents the general tendency of a set of numbers or data set. Like the
mean, median, and mode, it is a measure of central tendency. It is defined to be the weighted average of the
median and upper and lower quartiles.
Trimean = (Q1 + 2Q2 + Q3)/4
where Q1 And Q3 are the upper and lower quartiles (also known as hinges) and Q2 is the mean.

10. Example: find the trimean for the set 1, 2, 3, 4, 5, 6, 7.
Q1= {1,2,3}=2, Q2=4 , Q3= {5,6,7}=6
so, the trimean will be:
trimean={(4 x 2)+ 2 + 6} / 4
or, 16 / 4 = 4.
• Winsorized mean: Winsorized mean is a method of averaging that initially replaces the smallest and largest values
with the observations closest to them.After replacing the values, the arithmetic mean formula is then used to
calculate the winsorized mean.
winsorized mean=
𝑥𝑛
...𝑥𝑛
+
1
+
𝑥𝑛
+
2
...
𝑥𝑛
𝑁
where:
n = The number of largest and smallest data
points to be replaced by the observation
closest to them
​N = Total number of data points
Example: Find the winsorized mean of the dataset={9.1,9.2,9.2,9.2,9.2,9.3,9.9} [9.1 and 9.9 is replaced with
Winsorized mean= {9.2+9.2+9.2+9.2+9.2+9.3+9.3}/7= 9.22 the observations closest to them]

11. • Midhinge mean: In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of
location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary.The midhinge is related to the
interquartile range (IQR), the difference of the third and first quartiles, which is a measure of statistical dispersion.
The two are complementary in sense that if one knows the midhinge and the IQR, one can find the first and third
quartiles.
Midhinge=
𝑄1 +𝑄3
2
[Q1,Q3 are middle point of 1st and 3rd quarters]
Example: Find the midhinge mean of the dataset= { 2,4,6,8,10,12,14,16,18}
Q1= {2,4,6,8}=(4+6)/2 = 5
Q3={12,14,16,18}=(14+16)/2 = 15
According to the formula,
Midhinge mean=
5+15
2
=10.
• Quadratic mean: The quadratic mean (also called the root mean square*) is a type of average. It measures the
absolute magnitude of a set of numbers, and is calculated by:
• Squaring each number,
• Finding the mean of these squares,
• Taking the square root of that average.

12. The quadratic mean is equal to the square root of the mean of the squared values. The formula is:
quadratic mean=
𝒙𝟐
𝒏
Example:
Find the Root Mean Square of 2, 4, 9, 10, and 12.
Step 1: Count the number of items.
N = 5.
Set this number aside for a moment.
Step 2: Square all of the numbers. 22,42,92,102, 122 = 4, 16, 81, 100, 144.
Step 3: Add the numbers from Step 2 up: 4 + 16 + 81 + 100 + 144 = 345.
Step 4: Divide Step 3 (the sum) by Step 1 (number of items in the set):
345/5 = 69.
Step 5: Find square root of Step 4x √(69) = 8.31.

13. 2. What is Median?
• In statistics,Median is the middle value of the given list of data, when arranged in an order. The arrangement of
data or observations can be done either in ascending order or descending order.
• The median of a set of data is the middlemost number or center value in the set. The median is also the number
that is halfway into the set.
• To find the median, the data should be arranged, first, in order of least to greatest or greatest to the least value.

14. • For example, the median of 3,4,5,6,7;As because this dataset is odd dataset so the middle number will be the
median,so the median is 5.
Here is the other way from which we can find the median if the dataset is odd dataset ;So,the formula to calculate
the median is:
Median = {(n+1)/2}thterm
• For example, If there is an even number of observations, then there is no single middle value; the median is then
usually defined to be the mean of the two middle values: so the median of 3, 5, 7, 9 is (5+7)/2 = 6.
Here is the other way from which we can find the median if the dataset is Even dataset ;So,the formula to calculate
the median is:
Median=[(n/2)th term + {(n/2)+1}th]/2

15. 3.What is Mode?
• In statistics, the mode is the value that is repeatedly occurring in a given set. We can also say that the value or
number in a data set, which has a high frequency or appears more frequently, is called mode or modal value.
• For example, the mode of the set {3, 7, 8, 8, 9}, is 8. Therefore, for a finite number of observations, we can easily
find the mode. A set of values may have one mode or more than one mode or no mode at all.
• Mode formula for Ungrouped Data: To find the mode for ungrouped data, it would be better to arrange the data
values either in ascending or descending order, so that we can easily find the repeated values and their frequency.
Hence, the observation with the highest frequency will be the mode of the given data.
Example: The following table represents the number of wickets taken by a bowler in 10 matches. Find the mode of
the given set of data.
1 2 3 4 5 6 7 8 9 10
2 1 1 3 2 3 2 2 4 1
Match No.
No. of Wickets

16. It can be seen that 2 wickets were taken by the bowler frequently in different matches. Hence, the mode of the
given data is 2.
• Mode Formula For Grouped Data: In the case of grouped frequency distribution, calculation of mode just by
looking into the frequency is not possible. To determine the mode of data in such cases we calculate the modal
class. Mode lies inside the modal class. The mode of data is given by the formula:
Mode= l + (
𝑓1
− 𝑓0
2𝑓1
− 𝑓0
−𝑓2
) x h
Where,
l = lower limit of the modal class
h = size of the class interval
f1 = frequency of the modal class
f0 = frequency of the class preceding the modal class
f2 = frequency of the class succeeding the modal class

17. Example:In a class of 30 students marks obtained by students in mathematics out of 50 is tabulated as below.
Calculate the mode of data given.
The maximum class frequency is 12 and the class interval corresponding to this frequency is 20 – 30. Thus, the
modal class is 20 – 30.
Lower limit of the modal class (l) = 20
Size of the class interval (h) = 10
Frequency of the modal class (f1) = 12
Frequency of the class preceding the modal class (f0) = 5
Frequency of the class succeeding the modal class (f2)= 8
Marks Obtained Numbers of the students
10-20 5
20-30 12
30-40 8
40-50 5

18. • So,according to the formula,
Mode= 20 + (
12 − 5
2 𝑋 12 − 5 − 8
) x 10 = 26.364
Types of Mode:The different types of Mode are Unimodal, Bimodal, Trimodal, and Multimodal.
Unimodal Mode - A set of data with one Mode is known as a Unimodal Mode.
For example, the Mode of data set A = { 14, 15, 16, 17, 15, 18, 15, 19} is 15 as there is only one value repeating itself.
Hence, it is a Unimodal data set.
Bimodal Mode - A set of data with two Modes is known as a Bimodal Mode. This means that there are two data
values that are having the highest frequencies.
For example, the Mode of data set A = { 8,13,13,14,15,17,17,19} is 13 and 17 because both 13 and 17 are repeating
twice in the given set. Hence, it is a Bimodal data set.
Trimodal Mode - A set of data with three Modes is known as a Trimodal Mode. This means that there are three
data values that are having the highest frequencies.
For example, the Mode of data set A = {2, 2, 2, 3, 4, 4, 5, 6, 5,4, 7, 5, 8} is 2, 4, and 5 because all the three values are
repeating thrice in the given set. Hence, it is a Trimodal data set.
Multimodal Mode - A set of data with four or more than four Modes is known as a Multimodal Mode.
For example, The Mode of data set A = {100, 80, 80, 95, 95, 100, 90, 90,100 ,95 } is 80, 90, 95, and 100 because both
all the four values are repeated twice in the given set. Hence, it is a Multimodal data set.

19. • Besides, if any dataset is like doesn’t have any repeated values means no mode but still we want to know what is
the mode of that dataset,so there is one formula,
Mean - Mode = 3 (Mean - Median)
[This formula is also applicable when mean ≠ median,then we can calculate the mode value.]
Example-1: The dataset= [2,4,6,8,10,12,14,16,18] (There are no repeated values)
According to the formula,we have to calculate mean and median first.
Then,we put the numbers in the formula.
 The mean value is=
2 ,4 ,6,8,10,12,14,16,18
9
= 10.
 The median value is = 10
 As because mean=median so, the mode value will be 10.
Example-2: The dataset= [9,10,13,14,14,17,17,20]
 The mean value is=
9,10,13,14,14,17,17,20
8
= 14.25.
 The median value is=
14+14
2
= 14.
 The mode value is,
(mean-mode)=3(mean-median) Or, (14.25-mode)=3(14.25-14) Or, mode=13.50

20. 4.What is Midrange?
• In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample (statistics) defined as the
arithmetic mean of the maximum and minimum values which is divided by 2 :
Midrange=
𝑀𝑎𝑥.𝑣𝑎𝑙𝑢𝑒 + 𝑀𝑖𝑛.𝑣𝑎𝑙𝑢𝑒
2
• Example: Suppose, The dataset is = {2, 4, 7, 10, 14, 35}
Add together the greatest value and the least value and divide by 2.
The formula,
Midrange=
35+2
2
=18.5