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 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 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 let’s 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 Geometric Mean will be

\begin{align*}
GM &= (32\times 36 \times 36 \tmies 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 geometric mean considers how things are multiplied over time, rather than simply added.

Use and Application of Geometric Mean

Geometric Mean 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: It 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.

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