Session 07-01 - Descriptive Statistics Essentials

Section 07: Probability & Statistics

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Review from Section 06

Test your understanding of Integration

Find \(\int x \cdot e^x \, dx\) using integration by parts.
Evaluate \(\int_0^1 (2x + 1) \, dx\)
A company’s marginal profit is \(MP(x) = 60 - 2x\). Find the profit function if \(P(0) = -100\).
Find the area between \(y = x\) and \(y = x^2\) from \(x = 0\) to \(x = 1\).

Welcome to Probability & Statistics!

New Section Overview

Section 07 covers essential exam topics:

Session 07-01: Descriptive Statistics (today)
Session 07-02: Basic Probability Concepts
Session 07-03: Combinatorics & Counting
Session 07-04: Conditional Probability
Session 07-05: Bayes’ Theorem
Session 07-06: Contingency Tables
Session 07-07: Binomial Distribution
Session 07-08: Mock Exam 2

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Probability accounts for approximately 25% of the Feststellungsprüfung!

Learning Objectives

What You’ll Master Today

Calculate measures of central tendency: mean, median, mode
Compute measures of spread: range, variance, standard deviation
Interpret data distributions using histograms and box plots
Work with frequency distributions and relative frequencies
Apply statistical concepts to business scenarios

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This is foundational material - brief coverage to prepare for probability!

Part A: Measures of Central Tendency

The Three Averages

How do we summarize a data set with a single number?

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Three Measures of Center

Mean (Mittelwert): \(\bar{x} = \frac{\sum x_i}{n}\)
Median (Zentralwert): Middle value when data is sorted
Mode (Modalwert): Most frequently occurring value

Example: Sales Data

Monthly sales (in thousands €) for a store:

\[12, 15, 14, 18, 15, 22, 15, 16, 14, 19\]

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Mean: \[\bar{x} = \frac{12 + 15 + 14 + 18 + 15 + 22 + 15 + 16 + 14 + 19}{10} = \frac{160}{10} = 16\]

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Median: Sort: \(12, 14, 14, 15, 15, 15, 16, 18, 19, 22\)

Middle values: \(\frac{15 + 15}{2} = 15\)

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Mode: \(15\) (appears 3 times)

When to Use Each Measure

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Mean: Best for symmetric data without outliers
Median: Best for skewed data or data with outliers
Mode: Best for categorical data

Part B: Measures of Spread

How Spread Out Is the Data?

Two datasets can have the same mean but different spreads:

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Dataset A: \(48, 49, 50, 51, 52\) (mean = 50)

Dataset B: \(10, 30, 50, 70, 90\) (mean = 50)

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We need measures to quantify this difference!

Range

Simplest measure of spread:

\[\text{Range} = \text{Maximum} - \text{Minimum}\]

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Dataset A: Range \(= 52 - 48 = 4\)

Dataset B: Range \(= 90 - 10 = 80\)

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Range only uses two values - sensitive to outliers!

Variance and Standard Deviation

Variance (Varianz)

Population variance: \[\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}\]

Sample variance: \[s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}\]

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Standard Deviation (Standardabweichung)

\[\sigma = \sqrt{\sigma^2} \quad \text{or} \quad s = \sqrt{s^2}\]

Calculation Example

Data: \(4, 8, 6, 5, 3, 2, 8, 9, 2, 5\) (n = 10)

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Step 1: Calculate mean \[\bar{x} = \frac{4+8+6+5+3+2+8+9+2+5}{10} = \frac{52}{10} = 5.2\]

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Step 2: Calculate deviations squared \[(4-5.2)^2 + (8-5.2)^2 + ... = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 + 10.24 + 7.84 + 14.44 + 10.24 + 0.04 = 57.6\]

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Step 3: Variance and SD \[s^2 = \frac{57.6}{9} = 6.4 \quad \Rightarrow \quad s = \sqrt{6.4} \approx 2.53\]

Part C: Frequency Distributions

Organizing Data

Raw data: Test scores of 30 students

\[65, 72, 78, 81, 65, 73, 85, 92, 78, 72, 65, 88, 91, 73, 78, 82, 76, 72, 85, 78, 65, 73, 82, 79, 88, 73, 78, 85, 92, 78\]

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Question: How can we summarize this data effectively?

Frequency Table

Score Range	Frequency	Relative Frequency
60-69	4	4/30 = 13.3%
70-79	14	14/30 = 46.7%
80-89	9	9/30 = 30.0%
90-99	3	3/30 = 10.0%
Total	30	100%

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Relative frequency = Frequency / Total = Probability interpretation!

Histogram Visualization

Break - 10 Minutes

Part D: Box Plots (Five-Number Summary)

The Five-Number Summary

Five-Number Summary

Minimum (Min)
First Quartile (Q1) - 25th percentile
Median (Q2) - 50th percentile
Third Quartile (Q3) - 75th percentile
Maximum (Max)

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Interquartile Range (IQR): \(\text{IQR} = Q3 - Q1\)

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IQR contains the middle 50% of the data!

Box Plot Visualization

Detecting Outliers

Outliers are values that fall outside:

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\[\text{Lower fence: } Q1 - 1.5 \times \text{IQR}\] \[\text{Upper fence: } Q3 + 1.5 \times \text{IQR}\]

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Example: If \(Q1 = 65\), \(Q3 = 85\), then IQR \(= 20\)

Lower fence: \(65 - 1.5(20) = 35\)
Upper fence: \(85 + 1.5(20) = 115\)

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Any value below 35 or above 115 would be an outlier.

Part E: Business Applications

Quality Control Example

A factory measures the diameter of manufactured bolts (in mm):

\[10.2, 10.1, 10.0, 10.3, 9.9, 10.1, 10.0, 10.2, 10.1, 10.0\]

Target: 10.0 mm with tolerance ±0.3 mm

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Calculate:

Mean: \(\bar{x} = 10.09\) mm
Standard deviation: \(s = 0.12\) mm

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If we assume normal distribution, approximately 99.7% of bolts will be within \(\bar{x} \pm 3s = 10.09 \pm 0.36\) mm, which is within tolerance!

Sales Analysis Example

Weekly sales data for 8 weeks (in €1000):

\[45, 52, 48, 55, 62, 50, 48, 56\]

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Measure	Value	Interpretation
Mean	€52,000	Average weekly sales
Median	€51,000	Typical week
Std Dev	€5,300	Sales variability
Range	€17,000	Max spread

Guided Practice - 15 Minutes

Practice Problems

Work in pairs

Problem 1: Customer wait times (minutes): \(3, 5, 2, 8, 4, 6, 3, 7, 2, 10\)

Calculate mean, median, and mode
Calculate variance and standard deviation
Is the mean or median a better measure of center? Why?

Problem 2: Create a frequency table for exam scores: \(75, 82, 91, 78, 85, 68, 73, 88, 95, 79, 82, 76, 84, 90, 77\)

Connection to Probability

From Statistics to Probability

Key connection:

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\[\text{Relative Frequency} \approx \text{Probability}\]

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Example: If 30% of customers wait more than 5 minutes, then the probability that a randomly selected customer waits more than 5 minutes is approximately 0.30.

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This is the frequentist interpretation of probability - probability equals long-run relative frequency!

Wrap-Up & Key Takeaways

Today’s Essential Concepts

Mean, median, mode measure center differently
Variance and standard deviation measure spread
Box plots show distribution shape and outliers
Relative frequency connects to probability
Choose the right measure based on data characteristics

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Coming Next

Session 07-02: Basic Probability Concepts - sample spaces, events, and probability rules!

Homework Assignment

Tasks 07-01

Calculate descriptive statistics for business datasets
Interpret measures in context
Create and interpret frequency distributions
Prepare for probability concepts

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This material is foundational - make sure you’re comfortable before moving to probability!