Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, Python, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

Median of Ungrouped Data

by

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)		1st2nd3rd4th5th6th7th8th9th10th11th
Data Values81899096100102103104108109118
(Descending Order)1110987654321

Since the number of observations is odd ($n = 11$), the central value after arranging in ascending order will be the 6th value. and the 6th 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.12345678910
Data Values81100961089010210410310989

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

Observations No.1st2nd3rd4th5th6th7th8th9th10th
x81899096100102103104108109

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

Median of ungrouped data

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.

https://rfaqs.com

https://gmstat.com

Statistical Inference: An Introduction

by

Introduction to Statistical Inference

Inference means conclusion. When we discuss statistical inference, it is the branch of Statistics that deals with the methods to make conclusions (inferences) about a population (called reference population or target population), based on sample information. The statistical inference is also known as inferential statistics. As we know, there are two branches of Statistics: descriptive and inferential.

Statistical inference is a cornerstone of many fields of life. It allows the researchers to make informed decisions based on data, even when they can not study the entire population of interest. The statistical inference has two fields of study:

Statistical Inference

Estimation

Estimation is the procedure by which we obtain an estimate of the true but unknown value of a population parameter by using the sample information that is taken from that population. For example, we can find the mean of a population by computing the mean of a sample drawn from that population.

Estimator

The estimator is a statistic (Rule or formula) whose calculated values are used to estimate (a wise guess from data information) is used to estimate a population parameter $\theta$.

Estimate

An estimate is a particular realization of an estimator $\hat{\theta}$. It is the notation of a sample statistic.

Types of Estimators

An estimator can be classified either as a point estimate or an interval estimate.

Point Estimate

A point estimate is a single number that can be regarded as the most plausible value of the $\theta$ (notation for a population parameter).

Interval Estimate

An interval estimate is a set of values indicating confidence that the interval will contain the true value of the population parameter $\theta$.

Testing of Hypothesis

Testing of Hypothesis is a procedure that enables us to decide, based on information obtained by sampling procedure whether to accept or reject a specific statement or hypothesis regarding the value of a parameter in a Statistical problem.

Note that since we rely on samples, there is always some chance our inferences are not perfect. Statistical inference acknowledges this by incorporating concepts like probability and confidence intervals. These help us quantify the uncertainty in our estimates and test results.

Important Considerations about Testing of Hypothesis

  • Hypothesis testing does not prove anything; it provides evidence for or against a claim.
  • There is always a chance of making errors (Type I or Type II).
  • The results are specific to the chosen sample and significance level.

FAQs about Statistical Inference

  1. Define the term estimation.
  2. Define the term estimate.
  3. Define the term estimator.
  4. Write a short note on statistical inference.
  5. What is statistical hypothesis testing?
  6. What is the estimation in statistics?
  7. What are the types of estimations?
  8. Write about point estimation and intervention estimation.

https://rfaqs.com, https://gmstat.com

Multiple Regression Analysis

by

Introduction to Multiple Regression Analysis

Francis Galton (a biometrician) examines the relationship between fathers’ and sons’ height. He analyzed the similarities between the parent and child generation of 700 sweet peas. Galton found that the offspring of tall parents tended to be shorter and offspring of shorter parents tended to be taller. The height of the children depends ($Y$) upon the height of the parents ($X$). In case, there is more than one independent variable (IV), we need multiple regression analysis (MRA), also called multiple linear regression (MLR).

Multiple Linear Regression Model

The linear regression model (equation) for two independent variables (regressors) is

$$Y_{ij} = \alpha + \beta_1 X_{1i} + \beta_2 X_{2i} + \varepsilon_{ij}$$

The general linear regression model (equation) for $k$ independent variables is

$$Y_{ij} = \alpha + \beta_1 X_{1i} + \beta_2 X_{2i} + \beta_3X_{3i} + \cdots + \varepsilon_{ij}$$

The $\beta$s are all regression coefficients (partial slopes) and the $\alpha$ is the intercept.

The sample linear regression model is

$$\hat{y} = \hat{\alpha} + \hat{\beta}_1 x_{1i} + \hat{\beta}_2x_{2i} + \hat{\varepsilon}_{ij}$$

Multiple Regression Coefficients Formula

To fit the MLR equation for two variables, one needs to compute the values of $\hat{\beta}_1, \hat{\beta}_2$, and $\alpha$.

Multiple Regression Analysis Partial Coefficient 1

The yellow part of the above formula is the (“sum of the product of 1st independent and dependent variables”) multiplied by the (“sum of the square of 2nd independent variable).

The red part of the above formula is the (“Sum of the product of 2nd independent and dependent variables”) multiplied by the (“sum of the product of two independent variables”).

The green part of the above formula is the (“sum of the square of 1st independent variable”) multiplied by the (“sum of the square of 2nd independent variable”).

The blue part of the above formula is the (“square of the sum of the product of two independent variables”).

The formula for 2nd regression coefficient is

Multiple Regression Analysis Partial Coefficient 1

In short, note that the $S$ stands for the sum of squares and the sum of products.

Multiple Linear Regression Example

Consider the following data about two regressors ($X_1, X_2$) and one regressand variable ($Y$).

$Y$$X_1$$X_2$$X_1 y$$X_2 y$$X_1 X_2$$X_1^2$$X_2^2$
301015300450150100225
2258110176402564
161012160192120100144
737214921949
1421028140204100
8930526191007351238582

\begin{align*}
S_{x_1Y} &= \sum X_1 y – \frac{\sum X_1 \sum Y}{n} = 619 – \frac{30\times 59}{5} = 265\\
S_{x_1x_2} &= \sum X_1 X_2 – \frac{\sum X_1 \sum X_2}{n} = 351 – \frac{30 \times 52}{5} = 39\\
S_{X_1^2} &= \sum X_1^2 – \frac{(\sum X_1)^2}{n} = 238 -\frac{30^2}{5} = 58\\
S_{X_2^2} &= \sum X_2^2 – \frac{(\sum X_2)^2}{n} = 582 – \frac{52^2}{5} = 41.2\\
S_{X_2 y} &= \sum X_2 Y – \frac{\sum X_2 \sum Y}{n} =1007 – \frac{52 \times 89}{5} = 81.4
\end{align*}

\begin{align*}
\hat{\beta}_1 &= \frac{(S_{X_1 Y})(S_{X_2^2}) – (S_{X_2Y})(S_{X_1 X_2}) }{(S_{X_1^2})(S_{X_2^2}) – (S_{X_1X_2})^2} = \frac{(265)(41.2) – (81.4)(39)}{(58)(41.2) – (39)^2} = 8.91\\
\hat{\beta}_2 &= \frac{(S_{X_2 Y})(S_{X_1^2}) – (S_{X_1Y})(S_{X_1 X_2}) }{(S_{X_1^2})(S_{X_2^2}) – (S_{X_1X_2})^2} = \frac{(81.4)(58) – (265)(39)}{(58)(41.2) – (39)^2} = -6.46\\
\hat{\alpha} &= \overline{Y} – \hat{\beta}_1 \overline{X}_1 – \hat{\beta}_2 \overline{X}_2\\
&=31.524 + 8.91X_1 – 6.46X_2
\end{align*}

Important Key Points of Multiple Regression

  • Independent variables (predictors, regressors): These are the variables that one believes to influence the dependent variable. One can have two or more independent variables in a multiple-regression model.
  • Dependent variable (outcome, response): This is the variable one is trying to predict or explain using the independent variables.
  • Linear relationship: The core assumption is that the relationship between the independent variables and dependent variable is linear. This means the dependent variable changes at a constant rate for a unit change in the independent variable, holding all other variables constant.

The main goal of multiple regression analysis is to find a linear equation that best fits the data. The multiple regression analysis also allows one to:

  • Predict the value of the dependent variable based on the values of the independent variables.
  • Understand how changes in the independent variables affect the dependent variable while considering the influence of other independent variables.

Interpreting the Multiple Regression Coefficient

https://rfaqs.com

https://gmstat.com

Geometric Mean

by

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

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$
32Log 32 = 1.5051
36log 36 = 1.5563
36log 36 = 1.5563
37log 37 = 1.5682
39log 39 = 1.5911
41log 41 = 1.6128
45log 45 = 1.6532
46log 46 = 1.6628
48log 48 = 1.6812
Total14.3870

\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:

ClassesFrequency
65 to 849
85 to 10410
105 to 12417
125 to 14410
145 to 1645
165 to 1844
185 to 2045
Tota60

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

Classes$f$$X$$log\, X$$f \times log\, X$
65-84974.5log 74.5 = 1.872216.8494
85-1041094.5log 94.5 = 1.975419.7543
105-12417114.5log 114.5 = 2.058834.9997
125-14410134.5log 134.5 = 2.128721.2872
145-1645154.5log 154.5 = 2.188910.9446
165-1844174.5log 174.5 = 2.24188.9672
185-2045194.5log 194.5 = 2.288911.4446
Total60  124.2471

\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.

https://rfaqs.com

https://gmstat.com