The
mathematical formula used for calculating the arithmetic mean is:

**Mean = (Sum of values) / (Number of values) **

For example, if we have a dataset of 5 numbers: 10, 12, 14, 16, and 18, the arithmetic mean can be calculated as:

Mean = (10 + 12 + 14 + 16 + 18) / 5 = 14

So, the arithmetic mean of this dataset is 14

**Merits of Mean :**

Ø Arithmetic mean rigidly defined by Algebraic Formula.

Ø It is easy to calculate and is simple arithmetic to understand.

Ø It is based on all observations of the given data.

Ø It is capable of being treated mathematically hence it is widely used in statistical analysis.

Ø Arithmetic mean can be computed even if the derailed distribution is not known but some of the observation and number of the observation are known.

Ø It is least affected by the fluctuation of sampling.

Ø For every kind of data mean can be calculated.

**Demerits of Arithmetic Mean: **

Ø It can neither be determined by inspection or by graphical location.

Ø Arithmetic mean cannot be computed for qualitative data like data on intelligence, honesty and smoking habit etc.

Ø It is too much affected by extreme observations and hence it is not adequately represent data consisting of some extreme point.

Ø Arithmetic mean cannot be computed when class intervals have open ends.

Ø If any one of the data is missing then mean cannot be calculated.

**Let’s understand Merits of mean in detail with examples:**

The mean is a widely used measure of central tendency, and it has several merits that make it a useful tool for data analysis. Here are some examples of the merits of mean:

**Simplicity:** The mean is
a simple measure that is easy to calculate and interpret. It is calculated by
summing all of the values in a dataset and dividing by the number of values.
This makes it a useful tool for quick calculations and analysis.

**Example:**The mean
income of a group of employees can be easily calculated by adding up the income
of each employee and dividing by the total number of employees.

**Provides a single representative value:** The mean
provides a single value that represents the central tendency of the data. This
can be useful for comparing different datasets with other values and drawing
conclusions about the data.

**Example: **The mean
height of a group of individuals can be used to compare the heights of
different populations or to assess whether a particular group is taller or
shorter than average.

**Reflects the overall trend of the data:** The mean
reflects the overall trend of the data and is less affected by random
fluctuations or outliers in the data. This makes it a useful tool for
identifying trends and patterns in the data.

**Example: **The mean
score of a group of students on a test can be used to assess the overall
performance of the group, even if a few students scored significantly higher or
lower than the others.

**Useful for parametric statistical tests: **The mean is
a useful tool for parametric statistical tests, which assume that the data is
normally distributed. This makes it a useful tool for hypothesis testing and
statistical inference. The arithmetic mean is commonly used in statistical
analyses, such as regression analysis and hypothesis testing. It is a
well-established method of summarizing data and can provide insights into the
underlying patterns and trends in the data.

**Example: **Suppose we
want to study the relationship between a person's height and their weight. The
arithmetic mean can be used to calculate the average height and weight of a
sample of individuals, allowing us to analyze the relationship between the two
variables

**Example 2: **The mean of
a group of blood pressure readings can be used to assess whether a particular
treatment is effective at reducing blood pressure levels.

**Easy to calculate: **The
arithmetic mean is relatively easy to calculate and can be computed using a
simple formula. This makes it a quick and efficient method of summarizing large
datasets. For example,
suppose a class of 30 students took a test, and their scores are as follows:

80, 75, 90, 85, 95, 70, 60, 80, 85, 90, 75, 80, 90, 85, 80, 95, 90, 75, 85, 80, 70, 75, 80, 90, 85, 75, 85, 80, 75, 90

To calculate the mean score, we add up all the scores and divide by the number of students:

Mean score = (80 + 75 + 90 + ... + 90) / 30 = 82.5

**Represents the centre of the data:** The
arithmetic mean is a good representation of the centre of the data. It takes
into account all the values in the dataset and provides a single value that
summarizes all the observations data.

For example, suppose we want to compare the salaries of employees in two different departments of a company. The arithmetic mean can provide a clear and concise summary of the salaries in each department, allowing for easy comparison and analysis.

**Stable measure: **The
arithmetic mean is a stable measure of central tendency. It is not affected by
extreme values, as long as they are not too far from the rest of the data. This
makes it a reliable measure for many statistical applications.

**Preserves the original scale: **The mean
preserves the original scale of the data, which means that the units of
measurement are not lost. This is particularly useful when comparing data from
different sources or datasets.

**Widely used: **The
arithmetic mean is a commonly used measure of central tendency in many fields,
including finance, economics, science, and engineering. Its widespread use
makes it easy to compare and communicate results across different disciplines.

Overall, the merits of mean make it a useful tool for a wide range of applications, including data analysis, hypothesis testing, and statistical inference. Its simplicity, ability to provide a single representative value, and ability to reflect the overall trend of the data make it a valuable tool for researchers, analysts, and decision-makers.

**Demerits of mean with examples**

These examples demonstrate the practical implications of the demerits of mean. It is sensitive to outliers, not suitable for non-numerical data, misleading for skewed of data sets, not robust to changes in sample size, and not applicable for some distributions. Careful consideration of these demerits is important when selecting an appropriate measure of central tendency for a given dataset.

The mean has several demerits as a measure of central tendency, which limit its usefulness in certain applications. Here are some examples of the demerits of mean, along with their practical implications:

**Sensitive to outliers: **The mean is
sensitive to outliers, which can distort its value and provide an inaccurate
measure of central tendency. For example, suppose we have a dataset of income
levels for a population. If a small group of individuals has very high income
levels, the mean income for the population can be skewed upwards, providing an
inaccurate representation of the typical income level.

**Not suitable for non-numerical data: **The mean is
not suitable for non-numerical data, such as categorical or ordinal data. For
example, if we have a dataset of student grades that are categorized as A, B,
C, D, and F, the mean cannot be calculated, as these values are not numerical.

**Misleading for skewed data: **The mean can
be misleading for a data set that is skewed, where most of the values are
concentrated at one end of the distribution. For example, suppose we have a
dataset of salaries for a company. If most of the employees have low salaries,
but a few executives have very high salaries, the mean salary can be skewed
upwards, providing an inaccurate representation of the typical salary level.

**Not robust to changes in sample size: **The mean is
not robust to changes in sample size, which can affect its value significantly.
For example, suppose we have a dataset of exam scores for a class of 20
students, and the mean score is 75. If we add one more student who scored 100,
the mean score increases to 78.8, even though the new student's score is much
higher than the rest of the class.

**Not applicable for some distributions: **The mean is
not applicable for some distributions, such as bimodal or multi-modal
distributions, where the data has multiple peaks or modes. For example, suppose
we have a dataset of response times for a cognitive task, and the data has two
peaks, corresponding to two different processes. In this case, the mean cannot
be used to summarize the data, as it does not provide a representative measure
of central tendency for discrete data.

**Not always the most informative measure: **While the
mean provides a single representative value, it may not always be the best
measure most informative measure of central tendency. Other measures such as
the median or mode may be more appropriate in certain situations.

**Uses :**

- Mean is used in fields such as business, engineering and computer science.
- It is used in report card or in counting our population.
- In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent. For example, per capita GDP gives an approximation of the arithmetic average income of a nation's population.
- While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers.
- It’s used to compute the variance and SD (Standard Deviation). It’s good for inferential statistics.

**Why is merits and demerits of mean important?**

The merits and demerits of mean are important because they help us understand the strengths and limitations of this commonly used measure of central tendency. By understanding these factors, we can make informed decisions about when to use the mean and when to use alternative measures of central tendency.

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