Basic Statistics

Introduction to Median of Ungrouped Data

The post is about calculating the median ungrouped data. The median is the most central point (middlemost central value) of the data/set of observations, with the condition that the data or set of observations should be arranged in ascending or descending order. The median divides the data into two equal parts. That is the main objective of the median.

It is important to note that the criteria for finding the median for grouped and ungrouped data are different.

The primary and secondary data can be defined as:

1. Primary data, also called raw or ungrouped data, does not undergo any statistical procedure/method, which is not in the form of frequency distribution.
2. Secondary data may also be called group data if it is in the form of frequency distribution.

Let us discuss how to find the median for ungrouped data.

There are two cases for ungrouped data. These cases are based on no of observations which is $n$

When the number of observations is odd (Say $n$ i.e. $n$ is odd), and when the number of observations is even (Say $n$ i.e. $n$ is even).

Median Calculations

The data below contains the odd number of observations.

Observation No. (Ascending Order)	1st	2nd	3rd	4th	5th	6th	7th	8th	9th	10th	11th
Data Values	81	89	90	96	100	102	103	104	108	109	118
(Descending Order)	11	10	9	8	7	6	5	4	3	2	1

Since the number of observations is odd ($n = 11$), the central value after arranging in ascending order will be the 6^th value. and the 6^th value is 102. That is the median is 102 for the above data.

The position of the median can be located mathematically, as follows:

\begin{align*}
\tilde{x} &= \left( \frac{n+1}{2} \right)th\,\, \text{value}\\
&=\frac{11+1}{2} = 6th\,\, \text{value}
\end{align*}

The value at the 6th position (from sorted data) is 102. The $\tilde{x}$ can be read as “x-tild” which is the notation of the median.

Median for Even Numbers of Observations

Consider the following data that contains an even number of observations.

Observation No.	1	2	3	4	5	6	7	8	9	10
Data Values	81	100	96	108	90	102	104	103	109	89

Data after sorting (either in ascending or descending order) is

Observations No.	1st	2nd	3rd	4th	5th	6th	7th	8th	9th	10th
x	81	89	90	96	100	102	103	104	108	109

Since $n=10$ which is even, the central position (that is median) lies between the 5th value and the 6th value. This central value is the average of the 5th and 6th values (from the sorted data). The average of these two central observations is called the median. The two central positions are 100 and 102, take the average of these two numbers and find the median.

$$Median = \frac{100+102}{2} = 101$$

Median Formula for Large Data Sets

The median formula for large or small data sets can be represented mathematically.

• For large data sets one can find the median of data mathematically. The formula for both odd number of observations and even numbers of observations is different.

The point to remember when computing the median is that

• For an odd number of observations, the median is the centermost value after sorting the data
• For an even number of observations, the median is the average of two central values after sorting the data

\begin{align*}
\tilde{x} &= \frac{1}{2} \left[ \left(\frac{n}{2}th \, \, value \right)+ \left(\frac{n}{2}+1 \right)the \,\, value \right]\quad \quad \text{(When observations are even)}\\
&= \frac{n+1}{2} \quad \quad \text{(when observations are odd)}
\end{align*}

The other way of the median formula is

Consider, a data set containing 157 observations. To compute the median, first of all, you need to sort the data in either ascending or descending order. The formula for this data will be

$$\tilde{x} = \frac{n+1}{2} = \frac{157+1}{2}=79th$$.

The 79th observation in the sorted data will be the median of the data.

In case, if there are even number of observations (say $n=396$, the median will be

\begin{align*}
\tilde{x} &= \frac{1}{2}\left[\left(\frac{n}{2}\right)th + \left(\frac{n+1}{2}\right)th \right]\\
&=\frac{1}{2} \left[\frac{396}{2}th + \frac{396}{2}+1 \right]\\
&= \frac{1}{2} \left[198th + 199th\right]
\end{align*}

The average of 198th value and 199th value from the sorted data will be the median of the data.

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Introduction to the Geometric Mean

The geometric mean (GM) is a way of calculating an average, but instead of adding values like the regular (arithmetic) mean, it multiplies them and then takes a root. The geometric mean (a useful measure of central tendency) is defined as the $n$th root of the product of $n$ positive values.

If we have two observations, let’s say 9 and 4, then the geometric mean is the square root of the product of these values, which is 6 ($\sqrt{9\times 4}=6$. If there are three values, say  3, 9, and 3, then the geometric average will be the $sqrt[3]{3\times 9 \times 3} = 3$. In a similar pattern, mathematically, for $n$ number of observations ($x_1, x_2, \cdots, x_n$) then the Geometric Average Formula will be

$$GM = (x_1 \times x_2 \times x_3 \times \cdots \times x_n)^{\frac{1}{n} }$$

Geometric Mean Example

Suppose we have the following set of values: $x=32, 36, 36, 37, 39, 41, 45, 46, 48$. The Computation of the Geometric Mean will be

\begin{align*}
GM &= (32\times 36 \times 36 \times 37 \times 39 \times 41 \times 45 \times 46 \times 48)^{\frac{1}{9}}\\
&=(243790484520960)^{\frac{1}{9}} = 39.7
\end{align*}

For a large number of observations, one can compute the GM by taking the log of all observations using the following formula:

$$GM = antilog \left[\frac{\sum\limits_{i=1}^n log\, x}{n} \right]$$

$x$	$log\, x$
32	Log 32 = 1.5051
36	log 36 = 1.5563
36	log 36 = 1.5563
37	log 37 = 1.5682
39	log 39 = 1.5911
41	log 41 = 1.6128
45	log 45 = 1.6532
46	log 46 = 1.6628
48	log 48 = 1.6812
Total	14.3870

\begin{align*}
GM &= antilog \left[ \frac{\sum\limits_{i=1}^n log\, x}{n} \right]\\
&= antilog \left[\frac{14.3870}{9}\right] = antilog [1.5986]\\
&= 38.7
\end{align*}

One important point that should be remembered is that if any value in the data set is zero or negative, then the GM cannot be computed.

Geometric Mean for Grouped Data

The GM for grouped data can also be computed using the following formula:

$$GM = antilog \left[ \frac{\Sigma f\times log\, x}{\Sigma f} \right]$$

Suppose we have the following frequency distribution as follows:

Classes	Frequency
65 to 84	9
85 to 104	10
105 to 124	17
125 to 144	10
145 to 164	5
165 to 184	4
185 to 204	5
Tota	60

The GM of the above frequency distribution can be performed as follows

\begin{align*}
GM &= antilog \left[ \frac{124.2471}{60} \right]\\
&=antilog (2.0708) = 117.4
\end{align*}

The GM is particularly useful when dealing with rates of change or ratios, such as growth rates in investments. That is because the geometric mean considers how things are multiplied over time rather than simply added.

Use and Application of the Geometric Mean

The GM is useful in situations like:

• Investment returns: When one looks at average investment growth, one wants to consider how much one’s money is multiplied over time, not just the change each year. That is why the GM is better suited for this scenario.
• Rates of change: Similar to investment returns, if something is increasing or decreasing by a percentage each time, the GM is a more accurate measure of the overall change.
• Growth Rates: When dealing with percentages or ratios that change over time (like investment returns or population growth), the geometric mean provides a more accurate picture of the overall change compared to the arithmetic mean.
• Proportional Changes: This is helpful for situations where changes are multiplied, not added. For example, if a recipe calls for doubling all ingredients, the geometric mean of the original quantities represents the final amount.

Real-Life Examples

1. Finance (Average Investment Returns): To calculate the average rate of return on investments over time one can use the Geometric Mean? It is because the returns are compound that one cannot use the arithmetic mean. For example, Year 1 return = +10%, the Year 2 return = -20%, Year 3 return = +30%, the GM Return will give the true average annual return over 3 years.
2. Economics (Growth Rates): The GM should be used to compute average GDP growth, inflation, or population growth over multiple years. It is because growth over time is multiplicative. For example, GDP grows at 3%, 4%, and 5% over 3 years. The geometric mean provides the average annual growth rate.
3. Business (Average Rate of Change in Prices or Sales): To find the average percentage change in prices or sales across several periods, the GM can be used. For example, A product price increased by 10%, then decreased by 5%, then increased by 8%. The GM will give the true average percentage change.
4. Environmental Science (Air or Water Quality Data): The GM should be used to calculate the average concentration of pollutants, as environmental data often contains highly skewed values. For example, Pollution levels: 2, 4, 8, 50 → The arithmetic mean is skewed by 50, therefore, the Geometric Mean will give a better central tendency for such data.
5. Demographics (Fertility or Mortality Rates): In demographic research, to average birth or death rates across different countries or regions, The Geometric Mean should be used because these rates are often ratios and vary widely between groups.
6. Health & Medicine (Drug Dosage and Bacterial Growth): To measure average bacterial growth rates, enzyme activities, or dosage effectiveness. It is because these processes grow or decline exponentially.
7. Marketing (Average Performance Metrics): The Geometric Mean should be used to calculate the average conversion rate or engagement rate over multiple platforms or campaigns. Because metrics are often multiplicative percentages, and geometric mean gives a more accurate reflection.

Summary Table

Field	Application	Why Use Geometric Mean?
Finance	Investment returns	Captures compound interest effects
Economics	GDP, inflation, population growth	Works well with growth rates
Business	Sales/prices over time	Accurately represents percent changes
Environment	Pollution concentration levels	Handles skewed environmental data
Demographics	Birth/death rates	Compares ratios fairly
Medicine	Bacterial/drug growth rates	Models exponential biological change
Marketing	Engagement/conversion rates	Why Use the Geometric Mean?

Summary

The geometric mean is incredibly useful in real-life situations where values are multiplied together, grow exponentially, or vary in ratios or percentages — rather than being added. The GM is very useful, especially in business, finance, science, and data analysis.

FAQS about Geometric Mean

• What is meant by the Geometric Mean?
• In what situation should the GM be used?
• For what observations, the GM not computed?
• Write down the formula of GM for group and ungrouped data.
• Give some real-life examples that make use of the Geometric Mean.

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Introduction to Weighted Averages

The multipliers or sets of numbers that express more or less relative importance of various observations (data points) in a data set are called weights.

The weighted arithmetic mean (simply called weighted average or weighted mean) is similar to an ordinary arithmetic mean except that instead of each data point contributing equally to the final average, some data points contribute more than others. Weighted means are useful in a wide variety of scenarios. Weighted averages are used when there are a bunch of values, but some of those values are more important or contribute more to the overall result.

Example of Weighted Average

For example, a student may use a weighted mean to calculate his/her percentage grade in a course. In such an example, the student would multiply the weight of all assessment items in the course (e.g., assignments, exams, sessionals, quizzes, projects, etc.) by the respective grade that was obtained in each of the categories.

As an example, suppose in a course there are a total of 60 marks, while the distribution of marks is as follows, Assignment-1 has a weightage of 10%, Assignment-2 has a weightage of 10%, the mid-term examination has a weightage of 30% and the final term examination have the weightage of 50%. The scenario is described in the table below:

Assessment Item	Weight ($w_i$)	Grades ($x_i$)	Marks	Weighted Marks ($w_ix_i$)
Midterm	10 %	70 %	6	7 %
Assignment # 2	10 %	65 %	6	6.5 %
Midterm Examination	30 %	70 %	12	21 %
Final Term Examination	50 %	85 %	30	42.5 %
	100 %	290 %	60	77 %

Weighted Average Formula

Mathematically, the weighted average forma is given as

$$\overline{x}_w = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i}$$

Another Example

Consider another example: Suppose we have monthly expenditures of a family on different items with their quantity

Items	Weights ($w_i$)	Expenses ($x_i$)	Weighted Expenses $w_ix_i$
Food	7.5	290	2175
Rent	2.0	54	108
Clothing	1.5	96	144
Fuel and light	1.0	75	75
Misc	0.5	75	37.5
Total	12.5	590	2539.5

The average expenses will be: $AM = \frac{590}{5} = 118$.

However, the weighted average of the scenario will be $\overline{x}_w = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i} = \frac{2539.5}{12.5}=203.16$

Keeping in mind the importance of weight, the average monthly expenses of a family was 203.16, not 118.

Note that in a frequency distribution, the computation of relative frequency (rf) is also related to the concept of weighted averages.

Classes	Frequency	Mid point ($X$)	rf	Percentage
65-84	9	74.5	$\frac{9}{60} = 0.15$	15
85-104	10	94.5	$\frac{10}{60} = 0.17$	17
105-124	17	114.5	$\frac{10}{60} = 0.28$	28
125-144	10	134.5	$\frac{10}{60} = 0.17$	17
145-164	5	154.5	$\frac{5}{60} = 0.08$	8
165-184	4	174.5	$\frac{4}{60} =0.07$	7
185-204	5	194.5	$\frac{5}{60} =0.08$	8
Total	60

Some Real-World Examples of Weighted Averages

• Calculating class grade: Different assignments might have different weights (e.g., exams worth more than quizzes). A weighted mean considers these weights to determine the overall grade.
• Stock market performance: A stock index might use a weighted average to reflect the influence of large companies compared to smaller ones.
• Customer Satisfaction: Finding the average customer satisfaction score when some customers’ feedback might hold more weight (e.g., frequent buyers).
• Average Customer Spending: if some customers buy more frequently.
• Expected Value: Determining the expected value of outcomes with different probabilities.

The following are some important questions. What is the importance of weighted mean? Describe its advantages and disadvantages. What is an average? What are the qualities of a good average? What does Arithmetic mean? Describe the advantages and disadvantages of Arithmetic mean. In which situations do we apply arithmetic mean?

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