# Measures of Central Tendency

what is statistics ?

what is measure of central tendency.?

**1.Mean **

- Mean diagram image histogram
- Mean formula

**2.Median**

- Median definition
- Median meaning
- Median Formula
- Median calculate

**3.Mode**

- Mode definition
- Mode meaning
- Mode Formula
- Mode calculate

**What is statistics?**

Statistics (data) is defined as the science of figure. It is the science
of collecting, organizing, presenting, analyzing and interpretation the data.

Sir Ronald Fisher is the father of statistics.

Measure of
central tendency Correlation

Measure of
variability (Dispersion) Regression etc

Measure of
position

Skewness
& kurtosis

**Descriptive analysis statistics**: We simply summarize
the result from gathering data from a body, data group, or population and
reaching conclusions about that group.

**Inferential Statistics:** We draw inferences about the population from the
sample are generated from the process of gathering simple data from a group
,body, population and reaching conclusion about the larger group from which the
sample was drawn.

**Population: -** A population is the
total collection of objects that are of
interest in a statistical study.

**Sample**:
- A sample, being a subset, is typically smaller than the population. In a
statistical study, all elements of a sample are available for observation,
which is not typically the case for a population.

**Qualitative Data:-**** **Qualitative
data are measurements for which there is no natural numerical
scale, but which consist of attributes, label or other non numerical
characteristics.

**Quantitative Data**:- Quantitative
data are numerical measurements that arise from a natural numerical
scale.

**Levels of Measurements**

There are four different scales of
measurement. The data can be defined as being one of the four scales. The four
types of scales are:

**Nominal Scale**

A nominal scale is the 1st level
of measurement scale in which the numbers serve as “tags” or “labels” to
classify or identify the objects. A nominal scale usually deals with the
non-numeric variables or the numbers that do not have any value. It is
qualitative

**Example:**

An example of a nominal scale
measurement is given below:

M- Male

F- Female

**Ordinal Scale**

The ordinal scale is the second
level of measurement that reports the ordering and ranking of data without
establishing the degree of variation between them. Ordinal represents the
“order.” Ordinal data is known as qualitative data or categorical data. It can
be grouped, named and also ranked.

**Example:**

Ranking of school
students – 1st, 2nd, 3rd, etc.

Ratings in restaurants

Assessing the degree of
agreement

Totally agree

Agree

Neutral

Disagree

Totally disagree

**Interval Scale**

The interval scale is the third level
of measurement scale. It is defined as a quantitative measurement scale in
which the difference between the two variables is meaningful. In other words,
the variables are measured in an exact manner, not as in a relative way in
which the presence of zero is arbitrary.

**Example:**

Likert Scale

Net Promoter Score
(NPS)

Bipolar Matrix Table

**Ratio Scale**

The ratio scale is the fourth level
of measurement scale, which is quantitative. It is a type of variable
measurement scale. It allows researchers to compare the differences or
intervals. The ratio scale has a unique feature. It possesses the character of
the origin or zero points.

**Example:**

What is your weight in Kgs?

Less than 55 kgs

55 – 75 kgs

76 – 85 kgs

86 – 95 kgs

More than 95 kgs

**What is measure of central tendency?**

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. It sometimes called measures of central location. The mean (which is also called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.

The mean, median and mode are all valid measures of central tendency, but we can use under different conditions, some measures of central tendency become more appropriate to use than others.

**Mean**

It represents the average value of the dataset. It can be calculated as the sum of all the values in the dataset divided by the number of values. In general, it is considered as the arithmetic mean. Some other measures of mean used to find the central tendency are as follows:

1) Geometric Mean

2) Harmonic Mean

3) Weighted Mean

It is observed that if all the values in the dataset are the same, then all geometric, arithmetic and harmonic mean values are the same. If there is variability in the data, then the mean value differs. Calculating the mean value is completely easy. The formula to calculate the mean value is given as

**X1+X2+X3+X4+X5/N**

**Individual****Series**

**Example 1:2,3,4,5,6**

**Sol:**

**x¯¯¯=∑xN**

** **

**=2**+3+4+5+6/5

=20/5

=4

**Example:2-Find the mean from individual series.**

** x¯¯¯=∑xN**

**A.M=x1+x2+x3+x4+x5+x6+x7+x8/N**

20+25+30+15+20+40+60+54/8

=264/8

=33

**Discrete series**

**Example:3- Calculate mean from discrete series**

x | f |

5 | 3 |

10 | 7 |

15 | 5 |

20 | 12 |

25 | 9 |

**Sol:-**

x | f | x*f |

5 | 3 | 15 |

10 | 7 | 70 |

15 | 5 | 75 |

20 | 12 | 240 |

25 | 9 | 225 |

75 | 36 | 625 |

**After calculating all values,**

**x bar= Σfx/**

**Σf**

**=625/36**

**=17.36**

**Example:4-calculate mean from given series.**

x | f |

5 | 2 |

2 | 2 |

9 | 1 |

10 | 5 |

15 | 3 |

41 | 13 |

**Sol:-**

**x bar= Σfx/**

**Σf**

x | f | fx |

5 | 2 | 10 |

2 | 2 | 4 |

9 | 1 | 9 |

10 | 5 | 50 |

15 | 3 | 45 |

41 | 13 | 118 |

**x bar= Σfx/**

**Σf**

**Σfx=118**

**Σf=13**

**Put the values in the formula.**

**Continuous Series:-**

**Example:5-**

**Calculate mean from given continuous series.**

class | f |

10-20 | 4 |

20-30 | 10 |

30-40 | 26 |

40-50 | 8 |

50-60 | 2 |

Class | Mid value (x) | f | fx |

10-20 | 15 | 4 | 60 |

20-30 | 25 | 10 | 250 |

30-40 | 35 | 26 | 910 |

40-50 | 45 | 8 | 360 |

50-60 | 55 | 2 | 110 |

50 | 1690 |

**x bar= Σfx/**

**Σf**

**Mean is 33.8**

**Median**

Median is the middle value of the dataset in which the dataset is arranged in the __ascending order or in descending order__. When the dataset contains an even number of values, then the median value of the dataset can be found by taking the mean of the middle two values.

**Individual series.**

Median=

**Example:1**

Find the median for the following data set:

102, 56, 34, 99, 89, 101, 10, 54.**Sol-****Step 1:** Place the data in ascending order (smallest to highest).

10, 34, 54, 56, 89, 99, 101, 102.**Step 2: **Find the TWO numbers in the middle (where there are an equal number of data points above *and *below the two middle numbers).

10, 34, 54, 56, 89, 99, 101, 102**Step 3:** Add the two middle numbers and then divide by two, to get the average:

56 + 89 = 145

145 / 2 = 72.5.

**Discrete Series**

**Example:3 Calculate median in given discrete series.**

x | f |

5 | 2 |

2 | 2 |

9 | 1 |

10 | 5 |

15 | 3 |

41 | 13 |

x | f | c.f |

5 | 1 | 1 |

2 | 2 | 3 |

9 | 2 | 5 |

10 | 5 | 10 |

15 | 3 | 13 |

41 | 13 |

**Continuous series**

**Example:4**

Class | f |

10-20 | 4 |

20-30 | 10 |

30-40 | 26 |

40-50 | 8 |

50-60 | 2 |

### Sol:-

Class | Mid value(x) | f | c.f |

10-20 | 15 | 4 | 4 |

20-30 | 25 | 10 | 14 |

30-40 | 35 | 26 | 40 |

40-50 | 45 | 8 | 48 |

50-60 | 55 | 2 | 50 |

50 |

**For grouped data**

**(l+(n/2-c.f)/f*i**

### Mode

The mode represents the frequently occurring value in the dataset. Sometimes the dataset may contain multiple modes and in some cases, it does not contain any mode at all.

**Individual Series**

**Example:1 **

Find mode in given data set.

5, 4, 2, 3, 2, 1, 5, 4, 5

Consider the given dataset 5, 4, 2, 3, 2, 1, 5, 4, 5

Since the mode represents the most common value. Hence, the most frequently repeated value in the given dataset is 5.

**Discrete Series.**

**Example:2 ****Find the mode of given series.**

x | f |

5 | 2 |

2 | 2 |

9 | 1 |

10 | 5 |

15 | 3 |

**Continuous Series.**

**Example:3 Calculate mode in continuous series.**

Class | f |

10-20 | 2 |

20-30 | 2 |

30-40 | 1 |

40-50 | 5 |

50-60 | 3 |

Sol-

Mode for grouped data

=L+f1−f02f1−f0−f2×i

f = Frequency of modal class

= Frequency of class succeeding modal class

= Class interval.

=40+(5-1/2*5-1-3)*10

=46.6

**Example:4 Calculate mode in given series.**

Wages (in Rs.) | No. of workers |

0-5 | 3 |

05-10 | 7 |

10-15 | 15-f0 |

15-20 | 30-f1 |

20-25 | 20-f2 |

25-30 | 10 |

30-35 | 5 |

f0= Frequency of class preceding modal class

f = Frequency of modal class

= Frequency of class succeeding modal class

= Class interval.

**Ques.1 Find the missing frequency.**

A.M=17 yrs.

x (age) | f(no of persons) |

8 | 3 |

20 | 2 |

26 | ? |

29 | 1 |

Sol-

x (age) | f(no of persons) | fx |

8 | 3 | 24 |

20 | 2 | 40 |

26 | a=1 | 26a |

29 | 1 | 29 |

6+a | 93+26a |

**x̅ = Σfx/**

**Σf**

**A.M > G.M > H.M**

**Polled or combined Mean**

**X12bar =N1**x̅

**1+ N2**x̅

**2/N1+N2**

**X123 bar = N1 x̅**x̅

**1**+ N2x2+ N3**3/N1+N2+ N3**

**Ques.2**

**N1=8**

**x̅**

**1****=70 kg**

**N2=7**

**x̅**

**2 =55 kg**

**N3=5**

**x̅**

**3=71 kg**

**N1+N2+N3=20**

**X123 bar=?**

**Sol.**

**X123 bar**=

**N1 x̅**x̅

**1**+ N2x2+ N3**3/N1+N2+ N3**

**Ques.3- Given**

**N1+N2=Total no of student=150**

**N1=No of boys=?**

**N2= No of girls=?**

**x1 bar=70 kg**

**x2 bar=50 kg**

**x12 bar=60**

**x̅**=**12****N1 x̅**

**1**+ N2x2**/N1+N2**

**Ques.4**

**Ques.5**

**x̅**

**=**

**Σ**

**x/n**

**Σ**

**x=n**

**x̅**

**=25*52=1300**

**Σ**

**x=1300 (incorrect)**

**Correct**

**Σ**

**x=Incorrect**

**Σ**

**x-wrong obs.+correct obs**

**Based on the properties of the data, the measures of central tendency are selected.**

- All three measure of central tendency hold good incase you have symmetrical distribution of continuous data. But most of the time , the analyst uses the mean because it involves all the value in the distribution or dataset.

- If you have skewed distribution, the best measure of finding the central tendency is the median.

- If you have the original data, then both the median and mode are the best choice of measuring the central tendency.

- If you have categorical data, the mode is the best choice to find the central tendency.