Session 07-01 - Descriptive Statistics Essentials

Section 07: Probability & Statistics

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Review from Section 06

Test your understanding of Integration

  1. Find \(\int x \cdot e^x \, dx\) using integration by parts.

  2. Evaluate \(\int_0^1 (2x + 1) \, dx\)

  3. A company’s marginal profit is \(MP(x) = 60 - 2x\). Find the profit function if \(P(0) = -100\).

  4. 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

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

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?

Three Measures of Center

  1. Mean (Mittelwert): \(\bar{x} = \frac{\sum x_i}{n}\)

  2. Median (Zentralwert): Middle value when data is sorted

  3. Mode (Modalwert): Most frequently occurring value

Example: Sales Data

Monthly sales (in thousands €) for a store:

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

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

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

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

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

When to Use Each Measure

  • 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:

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

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

We need measures to quantify this difference!

Range

Simplest measure of spread:

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

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

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

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

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)

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

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

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

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

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%

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

  1. Minimum (Min)
  2. First Quartile (Q1) - 25th percentile
  3. Median (Q2) - 50th percentile
  4. Third Quartile (Q3) - 75th percentile
  5. Maximum (Max)

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

IQR contains the middle 50% of the data!

Box Plot Visualization

Detecting Outliers

Outliers are values that fall outside:

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

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

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

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

Calculate:

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

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):

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

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): \(3, 5, 2, 8, 4, 6, 3, 7, 2, 10\)

  1. Calculate mean, median, and mode
  2. Calculate variance and standard deviation
  3. 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:

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

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.

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

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

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