Applied Statistics using Python MCQ and 2 marks, Quizzes of Applied Statistics

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ASP LAB

Applied Statistical Programming

Complete Learning Guide

Experiments 1 – 9 | Python | NumPy | SciPy | Scikit-learn

Beginner-Friendly • Dark Mode • Viva Q&A Included

Table of Contents

Experiment 1 Matrix Operations using NumPy

Experiment 2 Mean, Variance and Standard Deviation

Experiment 3 Data Visualization – Univariate & Bivariate

Experiment 4 Correlation and Regression Analysis

Experiment 5 Hypothesis Testing – t-Test & Chi-Square

Experiment 6 Analysis of Variance (ANOVA)

Experiment 7 Principal Component Analysis (PCA)

Experiment 8 K-Means Clustering

Experiment 9 Time-Series Trend Analysis

B = np.array([[9, 8, 7], [6, 5, 4], [3, 2, 1]]) print('Matrix A:', A) print('Matrix B:', B)

Basic operations

print('Addition (A+B):', A + B) print('Subtraction (A-B):', A - B) print('Element-wise Multiplication:', A * B) print('Element-wise Division:', A / B) print('Matrix Multiplication (Dot):', np.dot(A, B)) print('Transpose of A:', A.T)

Statistics

print('Mean of A:', np.mean(A)) print('Median of A:', np.median(A)) print('Std Dev of A:', np.std(A)) print('Variance of A:', np.var(A)) print('Row-wise Mean:', np.mean(A, axis=1)) print('Column-wise Mean:', np.mean(A, axis=0)) print('Max:', np.max(A), ' Min:', np.min(A))

Normalization

norm_A = (A - np.min(A)) / (np.max(A) - np.min(A)) print('Normalized A:', norm_A)

Linear Algebra

det = np.linalg.det(A) print('Determinant:', det) if det != 0: print('Inverse:', np.linalg.inv(A)) else: print('Singular matrix – no inverse') eigenvalues, eigenvectors = np.linalg.eig(A) print('Eigenvalues:', eigenvalues) print('Eigenvectors:', eigenvectors)

Step-by-Step Procedure

• Step 1: Open your Python IDE (Spyder/Jupyter).
• Step 2: Import numpy as np at the top of your file.
• Step 3: Create Matrix A and Matrix B using np.array([[...]]).
• Step 4: Perform addition (A+B), subtraction (A-B), element-wise multiplication (A*B) and division

(A/B).

• Step 5: Use np.dot(A, B) for true matrix multiplication.
• Step 6: Use A.T to find the transpose.
• Step 7: Compute statistical values: np.mean(), np.median(), np.std(), np.var().
• Step 8: Normalize the matrix using the min-max formula.
• Step 9: Compute the determinant with np.linalg.det() and inverse with np.linalg.inv().

• Step 10: Compute eigenvalues and eigenvectors with np.linalg.eig().
• Step 11: Run the program and verify the output.

Observations

Operation Result Shape Key Function Addition A+B 3×3 A + B Subtraction A-B 3×3 A - B Element-wise Mult 3×3 A * B Matrix Mult 3×3 np.dot(A,B) Transpose 3×3 A.T Mean Scalar np.mean(A) Determinant Scalar np.linalg.det(A)

Result

The program successfully performs all matrix operations including addition, subtraction, element-wise and

dot-product multiplication, transpose, statistical calculations, normalization, determinant, inverse, and

eigenvalue decomposition using NumPy.

Conclusion

NumPy provides extremely efficient and concise tools for matrix operations that would otherwise require

many lines of manual code. These operations form the foundation of machine learning, image processing,

and engineering data analysis.

Real-World Applications

• Image processing (each image is a matrix of pixel values)
• Machine learning (weight matrices in neural networks)
• Computer graphics (transformation matrices for 3D rendering)
• Structural engineering (finite element analysis)
• Google PageRank algorithm (eigenvalue computation)

Viva Questions & Answers

Q: What is NumPy and why is it used?

A: NumPy (Numerical Python) is a library for numerical computing. It provides the ndarray object which

supports fast vectorized operations, making it much faster than Python lists for mathematical computations.

Q: Difference between a NumPy array and a Python list?

A: Python lists can hold mixed types and are slow for math. NumPy arrays hold a single data type, are stored

in contiguous memory, and support vectorized operations making them 50–100x faster.

Experiment 2 – Mean, Variance and Standard

Deviation

Aim

To write a Python program to compute mean, variance, and standard deviation of user-entered data using

NumPy.

Theory

These three measures are the most fundamental tools in descriptive statistics. They help summarize an

entire dataset with just a few numbers.

Mean (Average)

The mean is the sum of all values divided by the number of values. It represents the central value of the

dataset.

Mean ( μ ) = (x1 + x2 + ... + xn) / n

Variance

Variance measures how spread out the data is from the mean. A high variance means data points are

widely scattered; a low variance means they are clustered near the mean.

Variance ( σ ²) = Σ (xi - μ )² / n

Standard Deviation

Standard deviation is the square root of variance. It is in the same unit as the original data, making it more

interpretable.

Std Dev ( σ ) = √ Variance = √ ( Σ (xi - μ )² / n)

n Note: Standard deviation can NEVER be negative. It is always ≥ 0.

n Tip: If all values are identical, variance and std dev are both 0.

Materials Required

• Python 3.x
• NumPy library

Program

import numpy as np

Get number of elements from user

n = int(input('Enter number of elements: '))

Collect data values

data = [] for i in range(n):

value = float(input(f'Enter element {i+1}: ')) data.append(value)

Convert to NumPy array

data = np.array(data)

Compute statistics

mean = np.mean(data) variance = np.var(data) std_dev = np.std(data)

Display results

print('\nData:', data) print('Mean:', mean) print('Variance:', variance) print('Standard Deviation:', std_dev)

Step-by-Step Procedure

• Step 1: Import NumPy.
• Step 2: Ask the user how many elements they want to enter.
• Step 3: Loop n times and collect each value into a list.
• Step 4: Convert the list to a NumPy array using np.array().
• Step 5: Call np.mean(), np.var(), np.std() on the array.
• Step 6: Print all results clearly.

Sample Output

Enter number of elements: 5 Enter element 1: 10 Enter element 2: 20 Enter element 3: 30 Enter element 4: 40 Enter element 5: 50 Data: [10. 20. 30. 40. 50.] Mean: 30. Variance: 200. Standard Deviation: 14.

Observations

Input Data Mean Variance Std Dev 10, 20, 30, 40, 50 30.0 200.0 14. 5, 5, 5, 5, 5 5.0 0.0 0. 1, 100 50.5 2450.25 49.

Result

Experiment 3 – Data Visualization (Univariate &

Bivariate)

Aim

To write a Python program for data visualization for univariate and bivariate datasets using Matplotlib.

Theory

Data Visualization converts raw numbers into graphical representations, making patterns, trends, and

outliers instantly visible.

Univariate Data

Univariate data has only ONE variable (e.g., heights of students). We visualize its distribution using

histograms, box plots, or bar charts.

Bivariate Data

Bivariate data has TWO variables (e.g., study hours vs marks). We look at the relationship between them

using scatter plots and line plots.

Plot Types

• Histogram — shows frequency distribution of a single variable
• Scatter Plot — shows relationship between two variables as dots
• Line Plot — shows trend over ordered data (e.g., time)

n Note: Always label your axes and give a title. A plot without labels is incomplete.

Materials Required

• Python 3.x
• NumPy — for data arrays
• Matplotlib — for plotting (pip install matplotlib)

Program

import numpy as np import matplotlib.pyplot as plt

--- Univariate Dataset ---

data = np.array([10, 15, 20, 25, 30, 35, 40])

Histogram

plt.figure() plt.hist(data, bins=5, color='steelblue', edgecolor='white') plt.title('Univariate Data - Histogram') plt.xlabel('Values') plt.ylabel('Frequency') plt.show()

--- Bivariate Dataset ---

x = np.array([1, 2, 3, 4, 5]) y = np.array([2, 4, 6, 8, 10])

Scatter Plot

plt.figure() plt.scatter(x, y, color='orange') plt.title('Bivariate Data - Scatter Plot') plt.xlabel('X values') plt.ylabel('Y values') plt.show()

Line Plot

plt.figure() plt.plot(x, y, color='green', marker='o') plt.title('Bivariate Data - Line Plot') plt.xlabel('X values') plt.ylabel('Y values') plt.show()

Step-by-Step Procedure

• Step 1: Import numpy and matplotlib.pyplot.
• Step 2: Create univariate dataset as a NumPy array.
• Step 3: Plot histogram using plt.hist() with appropriate bins.
• Step 4: Add title and axis labels; show the plot.
• Step 5: Create bivariate dataset (x and y arrays).
• Step 6: Plot scatter chart using plt.scatter().
• Step 7: Plot line chart using plt.plot().
• Step 8: Add titles and labels to all plots.

Observations

Plot Type Data Type Function Used Purpose Histogram Univariate plt.hist() Show frequency distribution Scatter Bivariate plt.scatter() Show relationship between variables Line Bivariate plt.plot() Show trends over ordered data

Result

The program successfully visualizes univariate data using a histogram and bivariate data using scatter and

line plots.

Conclusion

Experiment 4 – Correlation and Regression

Analysis

Aim

To write a program to calculate correlation coefficient, generate a regression equation, and graph to

visualize and predict data trends.

Theory

Correlation

Correlation measures the strength and direction of the linear relationship between two variables. The

Pearson Correlation Coefficient (r) ranges from -1 to +1.

Value of r Meaning +1.0 Perfect positive correlation +0.7 to +0.9 Strong positive +0.4 to +0.6 Moderate positive 0 No correlation -0.7 to -0.9 Strong negative -1.0 Perfect negative correlation

Linear Regression

Regression finds the best-fit straight line through the data points. The line equation is y = mx + c where m

is slope and c is intercept.

Regression Line: y = mx + c

Slope: m = Σ (xi - x n )(yi - n ) / Σ (xi - x n )²

n Note: Correlation tells HOW STRONG the relationship is. Regression tells the EXACT EQUATION to

predict y from x.

Program

import numpy as np import matplotlib.pyplot as plt

Sample data

x = np.array([10, 20, 30, 40, 50]) y = np.array([15, 25, 35, 45, 55])

Correlation

correlation = np.corrcoef(x, y) print('Correlation Matrix:\n', correlation)

Regression coefficients y = mx + c

m, c = np.polyfit(x, y, 1) print(f'\nRegression Equation: y = {m:.2f}x + {c:.2f}')

Predicted values

y_pred = m * x + c

Plot

plt.figure() plt.scatter(x, y, color='orange', label='Actual Data') plt.plot(x, y_pred, color='cyan', label='Regression Line') plt.title('Regression Analysis') plt.xlabel('X values') plt.ylabel('Y values') plt.legend() plt.show()

Sample Output Correlation Matrix: [[1. 1.] [1. 1.]] Regression Equation: y = 1.00x + 5.

Observations X Values Y Values Correlation (r) Slope (m) Intercept (c) 10,20,30,40,50 15,25,35,45,55 1.0 1.0 5.

Result

The program calculates the correlation coefficient and regression equation and plots the best-fit line

through the data points.

Viva Questions & Answers

Q: What is correlation?

A: A statistical measure that expresses the extent to which two variables are linearly related. Ranges from -

to +1.

Q: What is Pearson correlation coefficient?

A: A number between -1 and +1 indicating the strength of linear relationship. +1 = perfect positive, -1 =

perfect negative, 0 = no relationship.

Q: What is regression?

A: A statistical method that models the relationship between variables and finds the best-fit line to predict one

variable from another.

Q: Difference between correlation and regression?

Experiment 5 – Hypothesis Testing (t-Test &

Chi-Square)

Aim

To conduct hypothesis testing using the T-Test (for numerical data) and the Chi-Square Test (for

categorical data) using Python.

Theory

Hypothesis Testing is a statistical method to make decisions about populations based on sample data.

We test two competing claims:

• Null Hypothesis (H n ) : No significant difference exists (default assumption)
• Alternative Hypothesis (H n ) : A significant difference does exist

We use the p-value to decide. If p-value < 0.05 (significance level α), we reject Hn.

t-Test

Used to compare the means of two groups. If the p-value is below 0.05, the groups are significantly

different.

t = (x nn - x nn ) / √ (s²/n n + s²/n n )

Chi-Square Test

Used for categorical data to check if two variables are independent. We compare observed frequencies

with expected frequencies.

χ ² = Σ (Observed - Expected)² / Expected

n Note: p-value < 0.05 → Reject H n (significant difference exists)

n Note: p-value ≥ 0.05 → Fail to reject H n (no significant difference)

Program — t-Test

from scipy import stats n = int(input('Enter number of elements in each group: ')) group1 = [float(input(f'Group1 value {i+1}: ')) for i in range(n)] group2 = [float(input(f'Group2 value {i+1}: ')) for i in range(n)] t_stat, p_value = stats.ttest_ind(group1, group2) print('T-statistic:', t_stat) print('P-value:', p_value) if p_value < 0.05: print('Reject H0: Significant difference exists.') else: print('Fail to reject H0: No significant difference.')

Program — Chi-Square Test

import numpy as np from scipy.stats import chi2_contingency rows = int(input('Enter number of rows: ')) cols = int(input('Enter number of columns: ')) observed = [] for i in range(rows): row = [int(input(f'Value at ({i+1},{j+1}): ')) for j in range(cols)] observed.append(row) observed_array = np.array(observed) chi2, p, dof, expected = chi2_contingency(observed_array) print('Chi-Square Value:', chi2) print('P-value:', p) print('Degrees of Freedom:', dof) print('Expected Frequencies:\n', expected) if p < 0.05: print('Reject H0: Variables are dependent.') else: print('Fail to reject H0: Variables are independent.')

Observations

Test Statistic p-value Decision t-Test -0.54 0.60 Fail to reject Hn Chi-Square 7.64 0.105 Fail to reject Hn

Result

Hypothesis testing was successfully conducted using Python for both the T-Test and Chi-Square Test with

correct interpretation of p-values.

Viva Questions & Answers

Q: What is hypothesis testing?

A: A statistical procedure to test claims about a population using sample data. We decide whether to reject

the null hypothesis based on the p-value.

Q: What is a null hypothesis?

A: Hn is the default claim — typically 'no difference exists'. We only reject it if the data provides strong

evidence against it.

Q: What is a p-value?

A: The probability of getting results as extreme as observed, assuming Hn is true. A small p-value (< 0.05)

means the result is unlikely under Hn.

Q: What is a t-test?

Experiment 6 – Analysis of Variance (ANOVA)

Aim

To perform Analysis of Variance (ANOVA) using Python to compare multiple datasets and determine

whether a statistically significant difference exists among their means.

Theory

ANOVA (Analysis of Variance) extends the t-test to compare three or more groups simultaneously.

Instead of performing multiple t-tests (which inflates error), ANOVA does it in one go.

How ANOVA Works

• It computes the F-statistic = Between-group variance / Within-group variance
• A large F-statistic suggests means are significantly different
• p-value < 0.05 means at least one group mean is different

F = (Between-Group Variance) / (Within-Group Variance)

Types of ANOVA

• One-Way ANOVA — one independent variable
• Two-Way ANOVA — two independent variables

n Note: ANOVA tells you THAT a difference exists, not WHICH groups differ. Use post-hoc tests (Tukey

HSD) for that.

Program

import numpy as np from scipy import stats

Sample datasets (scores from three groups)

group1 = np.array([85, 88, 90, 92, 86]) group2 = np.array([78, 80, 82, 79, 81]) group3 = np.array([90, 92, 94, 91, 93])

One-Way ANOVA

f_statistic, p_value = stats.f_oneway(group1, group2, group3) print('Group 1:', group1) print('Group 2:', group2) print('Group 3:', group3) print('\nF-Statistic:', f_statistic) print('P-Value:', p_value) if p_value < 0.05: print('\nResult: Significant difference exists between groups.') else: print('\nResult: No significant difference between groups.')

Sample Output

F-Statistic: 42. P-Value: 3.479e- Result: Significant difference exists between the group means.

Observations

Group Values Mean Group 1 85, 88, 90, 92, 86 88. Group 2 78, 80, 82, 79, 81 80. Group 3 90, 92, 94, 91, 93 92.

Result

The program successfully computes the F-statistic and p-value. Since p-value ≈ 3.48×10nn < 0.05, we

conclude that a significant difference exists between the group means.

Viva Questions & Answers

Q: What is ANOVA?

A: ANOVA (Analysis of Variance) is a statistical test that compares the means of three or more groups to

determine if at least one group mean is statistically different from others.

Q: Why is ANOVA used instead of multiple t-tests?

A: Using multiple t-tests increases the chance of Type I error (false positive). ANOVA controls this error rate

by testing all groups simultaneously.

Q: What is the F-statistic?

A: F = Between-group variance / Within-group variance. A large F means the group means differ more than

we'd expect by chance.

Q: What is the null hypothesis in ANOVA?

A: Hn: All group means are equal. Hn: At least one group mean is different.

Q: What is between-group variance?

A: The variability between the group means — how much the group averages differ from the overall average.

Q: What is within-group variance?

A: The variability of data points within each group around their own group mean. Also called residual or error

variance.

Q: What are the assumptions of ANOVA?

A: Normality (data in each group is normally distributed), Homogeneity of variance (equal variances across

groups), Independence of observations.

Q: Where is ANOVA applied in engineering?

A: Comparing material strengths from different manufacturers, testing different machine settings, quality

comparison across multiple production lines.