Presentation of Data in Statistics

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Since the primary data is in raw form or haphazard, it is not easy to examine the unorganized data. The scientist or researcher has organized the data in an understandable and meaningful way. In this post, we will learn about the organization/ presentation of data in Statistics. The presentation of data in statistics is a vital aspect, as it transforms raw data into meaningful and understandable information.

Classification/ Presentation of Data in Statistics

The classification is a widely used data organization technique which is further classified into three categories

  • Tabulation (Frequency Distribution and Contingency Tables)
  • Graphical Presentation of Data (Bar charts, Pie charts, scatter diagrams. line charts, etc.)
  • Textual Presentation of Data (Descriptive Statistics)

Classification of Data

Classification is defined as the process of dividing a set of data into different groups or categories so that they are homogeneous with respect to their characteristics and mutually exclusive. In other words, classification is a method that divides a set of data into different heterogeneous groups or sorts the data into different heterogeneous groups, by sort we mean a systematic arrangement of objects, individuals, and units in such a way that different categories are created.

The data can be classified/presented/organized in different ways, such as color classification, age classification, gender classification, and grade classification.

Tabulation

The classification of data in tabular form with suitable headings of tables, rows, and columns is called tabulation. There are different parts or components of a table: (i) Title, (ii) Column Caption, (iii) Row Caption, (iv) Footnotes, (v) Source note.

Presentation of Data in Statistics
  • Table Number: A number is allocated to the table for identification, particularly when there are a lot of tables in the study.
  • Title: The title of the table should explain what is contained in the table. The title must be concise, clear, brief, and set in bold type font on the top of the table. It may also indicate the time and place to which the data refer.
  • Stub or Row Designations: Each row of the table should be given a brief heading called stubs or stub items. For columns, it is called the stub column.
  • Column Headings or Captions: column designation is given on top of each column to explain to what the figures in the column refer. It should be concise, clear, and precise. This is called caption, or heading. Columns can also be numbered if there are four or more columns in a table.
  • Body of the Table: The data should be organized/ arranged in such a way that any data point/ figure can be located easily. Various types of numerical variables should be arranged in ascending order from left to right in rows and from top to bottom in columns. The columns and rows totals can also be given.
  • Source: At the bottom of the table, a note should be added indicating the primary and secondary sources from which data have been collected
  • Footnotes and references: If any item has not been explained properly, a separate explanatory note should be added at the bottom of the table.

Importance of Tabulation

In Tabulation, data are arranged and it makes data brief.

  • In tabulation, data is divided into various parts and for each part, there are totals and sub totals. Therefore, relationships between different parts can easily be established.
  • Since data is organized in a table with a title and a number, data can be easily identified and used for the required purpose.
  • Tables can be easily presented in the form of graphs.
  • Tabulations make complex data simple making it easy to understand the data.
  • Tabulation also helps in identifying mistakes and errors.
  • Tabulation condenses the collected data and it becomes easy to analyze the data from tables.
  • Tabulation saves time and costs as it is the easiest and most comprehensive method used to organize the data.
  • Since tabulation summaries, the large scattered data, the maximum information may be gained/collected from these tables.

Limitations of Tabulation

  • Tables contain only numerical data. The tables do not contain further details.
  • Qualitative expressions are not possible through tables.
  • Usually, tables are used by experts to conclude, but common men cannot understand them properly.

Examples of Tabulation

Consider, that a district is divided into two areas urban area and rural area, The Total population of the district is 271076 out of which only 46740 live in the urban area. The total male population of the district is 139699 and that of the urban area is 23083. The total unmarried population of the district is 112352 out of which 36864 are rural females. In the urban area unmarried people number 21072 out of which 12149 are males. Construct a table showing the population of the district by marital status, residence, and Gender.

Tabulation Presentation of Data in Statistics
Tabulation example presentation of data in statistics

Graphical Presentation of Data In Statistics

Visualization or Graphical presentation of data in statistics helps researchers visualize hidden information in a graphical/visual way. There are many types of graphical representations of the data:

  • Bar Charts: Bar charts are used to represent the frequency, percentage, or magnitude of different categories or groups in rectangular form. Simple bar charts are used to compare different categories while multiple bar charts are used to compare multiple categories over time or across groups. The stacked bar charts are used to show the composition of each category.
  • Pie Charts: Pie charts are used to represent the proportions of a whole as slices/sectors of a pie.
  • Line Graphs: Line graphs are used to show trends over time or relationships between variables.
  • Scatter plots: Scatter plots are used to visualize the relationship between two quantitative variables.
  • Histogram: Histograms are similar to bar charts where the bars are adjacent, representing the frequency distribution of a continuous variable.

Textual Presentation of Data in Statistics

Textual presentation of data includes descriptive statistics. Descriptive statistics summarizes the data using numerical measures like mean, median, mode, range, and standard deviation.

Selection of the Right Method for the Presentation of Data

For the presentation of data in statistics, one should be careful in selecting the right method of data representation. The selection or choice of the right method depends on:

  • Type of data: The visualization or textual presentation of data depends on the type of the data. For example, categorical data (such as gender, color, etc.) is often presented using bar charts or pie charts, while numerical data (such as age, marks, income, etc.) is better suited for histograms, line graphs, or scatter plots.
  • Purpose: To show the trends of data over time, one can use a line graph. A pie chart is suitable for comparing proportions. Therefore, the selection of presentation of data depends on the purpose, use, or application of data in real life.
  • Audience: The selection of different presentations of data depends on the familiarity of the audience with different types of graphs and charts. Simpler visualizations might be more effective for a general audience.

FAQS about Presentation of Data in Statistics

  1. What is meant by the presentation of data?
  2. What is the difference between tabulation, graphical presentation, and textual presentation of the data?
  3. What are the different parts of a table? explain in detail.
  4. Discuss different graphical representations.
  5. Discuss the selection of the right method depending on the type of data.
  6. What is the importance of tabulation in statistics?

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Eigenvalue Multicollinearity Detection

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In this post, we learn about the role of eigenvalue multicollinearity detection. In the context of the detection of multicollinearity, eigenvalues are used to assess the degree of linear dependence among explanatory (regressors, independent) variables in a regression model. Therefore, by understanding the role of eigenvalue multicollinearity detection, one can take appropriate steps to improve the reliability and interpretability of the regression models.

Decomposition of Eigenvalues and Eigenvectors

The pair-wise correlation matrix of explanatory variables is decomposed into eigenvalues and eigenvectors. Whereas Eigenvalues represent the variance explained by each Principal Component and Eigenvectors represent the directions of maximum variance.

The Decomposition Process

Firstly, compute the correlation coefficients between each pair of variables in the dataset.

Secondly, find the Eigenvalues and Eigenvectors: solve the following equation for each eigenvalue ($\lambda$) and eigenvector ($vV)

$$A v = \lambda v$$

where $A$ is the correlation matrix, $v$ is the eigenvector, and $\lambda$ is the eigenvalue.

The above equation essentially means that multiplying the correlation matrix ($A$) by the eigenvector ($v$) results in a scaled version of the eigenvector, where the scaling factor is the eigenvalue. This can be solved using various numerical methods, such as the power method or QR algorithm.

Interpreting Eigenvalue Multicollinearity Detection

A set of eigenvalues of relatively equal magnitudes indicates little multicollinearity (Freund and Littell 2000: 99). A small number of large eigenvalues suggests that a small number of component variables describe most of the variability of the original observed variables ($X$). Because of the score constraint, a number of large eigenvalues implies that there will be some small eigenvalues or some small variances of component variables.

A zero eigenvalue means perfect multicollinearity among independent/explanatory variables and very small eigenvalues imply severe multicollinearity. Conventionally, an eigenvalue close to zero (less than 0.01) or condition number greater than 50 (30 for conservative persons) indicates significant multicollinearity. The condition index, calculated as the ratio of the largest eigenvalue to the smallest eigenvalue $\left(\frac{\lambda_{max}}{\lambda_{min}}\right)$, is a more sensitive measure of multicollinearity. A high condition index (often above 30) signals severe multicollinearity.

Eigenvalue Multicollinearity Detection

The proportion of variances tells how much percentage of the variance of parameter estimate (coefficient) is associated with each eigenvalue. A high proportion of variance of an independent variable coefficient reveals a strong association with the eigenvalue. If an eigenvalue is small enough and some independent variables show a high proportion of variation with respect to the eigenvalues then one may conclude that these independent variables have significant linear dependency (correlation).

Presence of Multicollinearity in Regression Model

Since Multicollinearity is a statistical phenomenon where two or more independent/explanatory variables in a regression model are highly correlated, the existence/presence of multicollinearity may result in

  • Unstable Coefficient Estimates: Estimates of regression coefficients become unstable in the presence of multicollinearity. A small change in the data can lead to large changes in the estimates of the regression coefficients.
  • Inflated Standard Errors: The standard errors of the regression coefficients inflated due to the presence of multicollinearity, making it difficult to assess the statistical significance of the coefficients.
  • Difficulty in Interpreting Coefficients: It becomes challenging to interpret the individual effects of the independent variables on the dependent variable when they are highly correlated.

How to Mitigate the Effects of Multicollinearity

If multicollinearity is detected, several strategies can be employed to mitigate the effects of multicollinearity. By examining the distribution of eigenvalues, researchers (statisticians and data analysts) can identify potential issues and take appropriate steps to address them, such as feature selection or regularization techniques.

  • Feature Selection: Remove redundant or highly correlated variables from the model.
  • Principal Component Regression (PCR): Transform the original variables into a smaller set of uncorrelated principal components.
  • Partial Least Squares Regression (PLSR): It is similar to PCR but also considers the relationship between the independent variables and the dependent variable.
  • Ridge Regression: Introduces a bias-variance trade-off to stabilize the coefficient estimates.
  • Lasso Regression: Shrinks some coefficients to zero, effectively performing feature selection.
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MCQs Correlation and Regression Analysis 6

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The post is about MCQs correlation and Regression Analysis with Answers. There are 20 multiple-choice questions covering topics related to correlation and regression analysis, coefficient of determination, testing of correlation and regression coefficient, Interpretation of regression coefficients, and the method of least squares, etc. Let us start with MCQS Correlation and Regression Analysis with answers.

Online Multiple-Choice Questions about Correlation and Regression Analysis with Answers

1. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is

 
 
 
 

2. If the correlation coefficient ($r=1.00$) then

 
 
 
 

3. If the coefficient of determination is 0.49, the correlation coefficient may be

 
 
 
 

4. The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by

 
 
 
 

5. Which one of the following statements is true?

 
 
 
 

6. Testing for the existence of correlation is equivalent to

 
 
 
 

7. The correlation coefficient

 
 
 
 

8. If the correlation coefficient $r=1.00$ then

 
 
 
 

9. Which of the following does the least squares method minimize?

 
 
 
 

10. Which one of the following situations is inconsistent?

 
 
 
 

11. Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$

 
 
 
 

12. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is

 
 
 
 

13. What do we mean when a simple linear regression model is “statistically” useful?

 
 
 
 

14. The strength of the linear relationship between two numerical variables may be measured by the

 
 
 
 

15. The true correlation coefficient $\rho$ will be zero only if

 
 
 
 

16. In a simple linear regression problem, $r$ and $\beta_1$

 
 
 
 

17. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is

 
 
 
 

18. The slope ($b_1$) represents

 
 
 
 

19. If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a

 
 
 
 

20. The $Y$ intercept ($b_0$) represents the

 
 
 
 

MCQs Correlation and Regression Analysis with Answers

MCQs Correlation and Regression Analysis

  • The $Y$ intercept ($b_0$) represents the
  • The slope ($b_1$) represents
  • Which of the following does the least squares method minimize?
  • What do we mean when a simple linear regression model is “statistically” useful?
  • If the correlation coefficient $r=1.00$ then
  • If the correlation coefficient ($r=1.00$) then
  • Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$
  • Testing for the existence of correlation is equivalent to
  • The strength of the linear relationship between two numerical variables may be measured by the
  • In a simple linear regression problem, $r$ and $\beta_1$
  • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is
  • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is
  • If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a
  • The correlation coefficient
  • If the coefficient of determination is 0.49, the correlation coefficient may be
  • The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by
  • Which one of the following situations is inconsistent?
  • Which one of the following statements is true?
  • The true correlation coefficient $\rho$ will be zero only if
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MCQs Probability Quiz

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The post is about the Online MCQs Probability Quiz. There are 20 multiple-choice questions covering topics related to random experiments, random variables, expectations, rules of probability, events and types of events, and sample space. Let us start with the Probability Quiz.

Please go to MCQs Probability Quiz to view the test

MCQs Probability Quiz

Online MCQs Probability Quiz with Answers

  • Consider a dice with the property that the probability of a face with $n$ dots showing up is proportional to $n$. What is the probability of the face showing 4 dots?
  • Let $X$ be a random variable with a probability distribution function $$f (x) = \begin{cases} 0.2 & \text{for  } |x|<1 \ 0.1 & \text{for } 1 < |x| < 4\ 0 & \text{otherwise} \end{cases}$$ The probability P (0.5 < x < 5) is ————-
  • Runs scored by batsmen in 5 one day matches are 50, 70, 82, 93, and 20. The standard deviation is ————-.
  • Find the median and mode of the messages received on 9 consecutive days 15, 11, 9, 5, 18, 4, 15, 13, 17.
  • $E (XY)=E (X)E (Y)$ if $x$ and $y$ are independent.
  • Mode is the value of $x$ where $f(x)$ is a maximum if $X$ is continuous.
  • A coin is tossed up 4 times. The probability that tails turn up in 3 cases is ————–.
  • If $E$ denotes the expectation the variance of a random variable $X$ is denoted as?
  • $X$ is a variate between 0 and 3. The value of $E(X^2)$ is ————-.
  • The random variables $X$ and $Y$ have variances of 0.2 and 0.5, respectively. Let $Z= 5X-2Y$. The variance of $Z$ is?
  • In a random experiment, observations of a random variable are classified as
  • A number of individuals arriving at the boarding counter at an airport is an example of
  • If $A$ and $B$ are independent, $P(A) = 0.45$ and $P(B) = 0.20$ then $P(A \cup B)$
  • If a fair dice is rolled twice, the probability of getting doublet is
  • If a fair coin is tossed 4 times, the probability of getting at least 2 heads is
  • If $P(B) \ne 0$ then $P(A|B) = $
  • The collection of all possible outcomes of an experiment is called
  • An event consisting of one sample point is called
  • An event consisting of more than one sample point is called
  • When the occurrence of an event does not affect the probability of occurrence of another event, it is called
https://itfeature.com probability quiz with answers

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