Lecture 2: Descriptive Statistics I

Understanding Data (PSTAT 5A) – Week 1

Narjes Mathlouthi

2025-06-25

Today’s Agenda

  1. Introduction to Descriptive Statistics (10 minutes)
    • What is descriptive statistics?
    • Why it matters before any modeling
  2. Types of Data and Measurement Scales (15 minutes)
  3. Measures of Central Tendency (35 minutes)
    • Mean (12 minutes)
    • Median (12 minutes)
    • Mode (11 minutes)
  4. Python Implementation (15 minutes)
  5. Real-world Applications and Interpretation (5 minutes)

Learning Objectives (LOs)

By the end of this lecture you should be able to:

  • Define descriptive statistics and explain its importance in data analysis (Section 1)

  • Distinguish between different types of data and measurement scales (Section 2)

  • Calculate and interpret measures of central tendency (mean, median, mode)(Section 3)

  • Understand when to use each measure of central tendency

  • Apply Python to compute descriptive statistics(Section 9)

  • Interpret basic descriptive statistics in real-world contexts(Section 10)

Introduction to Descriptive Statistics

10 minutes

Statistics

Facts are stubborn, but statistics are more pliable.

Mark Twain

  • Statistics refers to the mathematics and techniques with which we understand data. 1

What are Descriptive Statistics?

Descriptive statistics are numerical and graphical methods used to summarize, organize, and describe data in a meaningful way.

Purpose of Descriptive Statistics

  • Summarize large datasets into manageable information
  • Identify patterns and trends in data
  • Communicate findings clearly to others
  • Prepare data for further analysis
  • Make initial assessments about data quality

Descriptive vs. Inferential Statistics

Descriptive Statistics Inferential Statistics
Describes what the data shows Makes predictions about populations
Summarizes sample data Uses sample data to make generalizations
No conclusions beyond the data Draws conclusions beyond the immediate data
Examples: mean, median, graphs Examples: hypothesis testing, confidence intervals

Descriptive vs. Inferential Statistics

Types of Data and Measurement Scales

15 minutes

Why Data Types Matter

Understanding data types is crucial because:

  • Different statistics are appropriate for different data types
  • Statistical methods depend on the level of measurement
  • Misapplying statistics can lead to incorrect conclusions

Types of Data

graph TD
    A[Variable Types]
    A --> B[Categorical]
    A --> C[Numerical]
    B --> B1["Nominal<br/>e.g., gender, major"]
    B --> B2["Ordinal<br/>e.g., rating, education level"]
    C --> C1["Discrete<br/>e.g., count (# of students)"]
    C --> C2["Continuous<br/>e.g., measurements like height, weight"]
    
    classDef root fill:#e1f5fe,stroke:#01579b,stroke-width:3px
    classDef categorical fill:#f3e5f5,stroke:#4a148c,stroke-width:2px
    classDef numerical fill:#e8f5e8,stroke:#1b5e20,stroke-width:2px
    classDef nominal fill:#fce4ec,stroke:#880e4f,stroke-width:2px
    classDef ordinal fill:#fff3e0,stroke:#e65100,stroke-width:2px
    classDef discrete fill:#e0f2f1,stroke:#00695c,stroke-width:2px
    classDef continuous fill:#f1f8e9,stroke:#33691e,stroke-width:2px
    
    class A root
    class B categorical
    class C numerical
    class B1 nominal
    class B2 ordinal
    class C1 discrete
    class C2 continuous

Categorical Data (Qualitative)

Nominal Data

  • Categories with no natural order
  • Examples: gender, color, brand names, marital status
  • Appropriate statistics: mode, frequency counts, proportions

Ordinal Data

  • Categories with a natural order or ranking
  • Examples: education level, satisfaction ratings, letter grades
  • Appropriate statistics: mode, median, percentiles

Numerical Data (Quantitative)

Discrete Data

  • Countable values, often integers
  • Examples: number of children, cars sold, defective items
  • Can take on finite or countably infinite values

Continuous Data

  • Can take any value within a range
  • Examples: height, weight, temperature, time
  • Measured rather than counted

Measurement Scales Summary

Scale Type Properties Examples Appropriate Statistics
Nominal Categorical Categories only Gender, Color Mode, Frequency
Ordinal Categorical Order matters Rankings, Grades Mode, Median
Interval Numerical Equal intervals, no true zero Temperature (°C) Mean, Median, Mode
Ratio Numerical Equal intervals, true zero Height, Weight, Income All statistics

Measures of Central Tendency

35 minutes

Overview


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graph TD
    A["Measures of Central Tendency"]
    A --> B["Mean (Average)"]
    A --> C["Median (Middle Value)"]
    A --> D["Mode (Most Frequent)"]
    
    B --> B1["Sum of all values ÷<br/>Number of values"]
    B --> B2["Best for:<br/>Symmetric distributions"]
    B --> B3["Affected by:<br/>Outliers"]
    
    C --> C1["Middle value when<br/>data is ordered"]
    C --> C2["Best for:<br/>Skewed distributions"]
    C --> C3["Resistant to:<br/>Outliers"]
    
    D --> D1["Value that appears<br/>most frequently"]
    D --> D2["Best for:<br/>Categorical data"]
    D --> D3["Can have: Multiple modes<br/>or no mode"]
    
    classDef main fill:#e3f2fd,stroke:#0d47a1,stroke-width:4px
    classDef mean fill:#fff3e0,stroke:#e65100,stroke-width:4px
    classDef median fill:#e8f5e8,stroke:#2e7d32,stroke-width:4px
    classDef mode fill:#f3e5f5,stroke:#7b1fa2,stroke-width:4px
    classDef details fill:#fafafa,stroke:#616161,stroke-width:4px
    
    class A main
    class B mean
    class C median
    class D mode
    class B1,B2,B3,C1,C2,C3,D1,D2,D3 details


What is Central Tendency?

Central tendency describes the center or typical value of a dataset.

It answers the question: “What is a representative value for this data?”

The Mean (Arithmetic Average)

12 minutes

Definition and Formula

🎯 Definition: The mean is the sum of all values divided by the number of values.

Formula

  • For a sample: \bar{x} = \frac{\sum x}{n}
  • For a population: \mu = \frac{\sum x}{N}

Where:

  • \bar{x} (x-bar) = sample mean
  • \mu (mu) = population mean
  • \sum x = sum of all values
  • n = sample size, N = population size

Example Calculation

Student test scores: 85, 90, 78, 92, 88

Mean = \frac{85 + 90 + 78 + 92 + 88}{5} = \frac{433}{5} = 86.6

Properties of the Mean

  • Uses all data points - every value affects the mean
  • Sensitive to outliers - extreme values can distort the mean
  • Unique - there is only one mean for a dataset
  • Can be calculated for interval and ratio data
  • Balancing point - sum of deviations from mean equals zero

When to Use the Mean

✅ Use the mean when:

  • Data is approximately symmetric
  • No extreme outliers present
  • Working with interval or ratio data
  • Need to use the value in further calculations

Advantages and Disadvantages

Advantages

  • Uses all information in the dataset
  • Algebraically defined and mathematically tractable
  • Widely understood and accepted

Disadvantages

  • Affected by outliers and skewed distributions
  • May not represent a typical value in skewed data
  • Cannot be used with nominal or ordinal data

The Median

12 minutes

Definition

🎯 Definition: The median is the middle value when data is arranged in ascending or descending order.

Calculation Steps

  1. Arrange data in ascending order
  2. Find the middle position:
    • If n is odd: position = \frac{n + 1}{2}
    • If n is even: average of positions \frac{n}{2} and \frac{n}{2} + 1

Examples

Odd number of values:

  • Data: 12, 15, 18, 20, 25

  • What is the median here ?

  • Median = 18 (middle value)

Even number of values:

  • Data: 10, 15, 20, 25, 30, 35

  • What is the median here ?

  • Median = \frac{20 + 25}{2} = 22.5

Properties of the Median

  • Not affected by outliers - resistant measure
  • Represents the 50th percentile
  • May not be an actual data value (when n is even)
  • Appropriate for ordinal, interval, and ratio data
  • Divides data into two equal halves

When to Use the Median

✅ Use the median when:

  • Data is skewed (not symmetric)
  • Outliers are present
  • Working with ordinal data
  • Want a robust measure of central tendency
  • Data represents income, housing prices, or similar distributions

Median vs Mean with Outliers

Consider household incomes: $30,000, $32,000, $35,000, $38,000, $2,000,000

  • Mean = $427,000 (not representative of typical household)
  • Median = $35,000 (better represents typical household)

The Mode

11 minutes

Definition

🎯 Definition: The mode is the value that appears most frequently in a dataset.

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graph TD
    A["Mode Types"]
    A --> B["Unimodal<br/>One peak"]
    A --> C["Bimodal<br/>Two peaks"]
    A --> D["Multimodal<br/>Multiple peaks"]
    A --> E["No Mode<br/>No repeated values"]
    
    classDef main fill:#e1f5fe,stroke:#01579b,stroke-width:3px,font-size:16px
    classDef types fill:#f5f5f5,stroke:#424242,stroke-width:2px,font-size:14px
    
    class A main
    class B,C,D,E types

Types of Distributions by Mode

  • Unimodal: One mode
  • Bimodal: Two modes
  • Multimodal: More than two modes
  • No mode: All values appear with equal frequency

Mode Examples

Example 1:

Data: 2, 3, 3, 4, 5, 5, 5, 6, 7

Analysis:

  • Count each value: 2(1), 3(2), 4(1), 5(3), 6(1), 7(1)
  • Most frequent value: 5 appears 3 times
  • Mode = 5
  • Type: Unimodal (one mode)

Example 2:

Data: 1, 2, 2, 3, 4, 4, 5

Analysis:

  • Count each value: 1(1), 2(2), 3(1), 4(2), 5(1)
  • Most frequent values: 2 and 4 both appear twice
  • Modes = 2 and 4
  • Type: Bimodal (two modes)

Example 3:

Data: 1, 2, 3, 4, 5

Analysis:

  • Count each value: 1(1), 2(1), 3(1), 4(1), 5(1)
  • All values appear exactly once
  • No mode (no value repeats)
  • Type: No mode

Mode for Different Data Types

Categorical Data:

Favorite colors: Red, Blue, Blue, Green, Blue, Red, Blue Mode = Blue

Continuous Data:

Often requires grouping into intervals or bins Example: Heights grouped into ranges

Properties of the Mode

  • Can be used with any type of data (nominal, ordinal, interval, ratio)
  • Not affected by outliers
  • May not exist or may not be unique
  • Represents the most common value
  • Easy to identify in frequency distributions

When to Use the Mode

✅ Use the mode when:

  • Working with categorical (nominal) data
  • Want to identify the most popular or common choice
  • Data has clear peaks in frequency
  • Quality control applications (most common defect type)
  • Business applications (best-selling product, most common customer complaint)

Comparing Measures of Central Tendency

Measure Best for Data Type Strengths Weaknesses Affected by Outliers?
Mean Interval, Ratio Uses all data, mathematically tractable Sensitive to outliers Yes
Median Ordinal, Interval, Ratio Robust to outliers, represents middle Ignores extreme values No
Mode All types Works with categorical, identifies most common May not exist/be unique No

Shape of Distribution Effects

  • Symmetric distribution: Mean ≈ Median ≈ Mode
  • Right-skewed (positively skewed): Mean > Median > Mode
  • Left-skewed (negatively skewed): Mode > Median > Mean

Python Implementation

15 minutes

import numpy as np # for data manipulation 
import pandas as pd # for data manipulation 
from scipy import stats # for advanced statistics
import matplotlib.pyplot as plt # for visulization
import seaborn as sns # for advanced visulization
Library Purpose Key Functions
NumPy Numerical computing np.mean(), np.median(), np.std()
Pandas Data manipulation df.describe(), df.mean(), df.median()
SciPy Scientific computing stats.mode(), stats.describe()
Matplotlib Basic plotting plt.plot(), plt.hist(), plt.boxplot()
Seaborn Statistical viz sns.histplot(), sns.boxplot()

Central Measures

# Import numpy library for numeric operations
import numpy as np 

# Import pandas library for data structures
import pandas as pd

# Define sample data as a list of test scores
data = [85, 90, 78, 92, 88, 91, 85, 87, 89, 86]

# Compute the arithmetic mean using NumPy
mean_np = np.mean(data)
print(f"Mean (NumPy): {mean_np:.2f}")

# Convert the data list into a pandas DataFrame
df = pd.DataFrame({'scores': data})

# Compute the arithmetic mean using Pandas
mean_pd = df['scores'].mean()
print(f"Mean (Pandas): {mean_pd:.2f}")

# Manually sum all scores and divide by count
manual_mean = sum(data) / len(data)
print(f"Mean (Manual): {manual_mean:.2f}")
Mean (NumPy): 87.10
Mean (Pandas): 87.10
Mean (Manual): 87.10
# Compute the median value using NumPy
median_np = np.median(data)
print(f"Median (NumPy): {median_np:.2f}")

# Compute the median value using Pandas
median_pd = df['scores'].median()
print(f"Median (Pandas): {median_pd:.2f}")

# Sort the data list in ascending order
sorted_data = sorted(data)

# Determine the number of elements
n = len(sorted_data)

# If even count, average the two middle elements; else take middle element
if n % 2 == 0:
    manual_median = (sorted_data[n//2 - 1] + sorted_data[n//2]) / 2
else:
    manual_median = sorted_data[n//2]

print(f"Median (Manual): {manual_median:.2f}")
Median (NumPy): 87.50
Median (Pandas): 87.50
Median (Manual): 87.50
# Import SciPy stats module to compute mode
from scipy import stats

# Use SciPy to find the most common value and its count
mode_result = stats.mode(data, keepdims=True)
print(f"Mode (SciPy): {mode_result.mode[0]}, Count: {mode_result.count[0]}")

# Use Pandas to get mode(s) from the DataFrame
mode_pd = df['scores'].mode()
print(f"Mode (Pandas): {mode_pd.values}")

# Import Counter for manual frequency counting
from collections import Counter

# Count occurrences of each score
counter = Counter(data)

# Find the highest frequency
max_count = max(counter.values())

# Identify all values that appear with that frequency
modes = [k for k, v in counter.items() if v == max_count]

print(f"Mode (Manual): {modes}, Count: {max_count}")
Mode (SciPy): 85, Count: 2
Mode (Pandas): [85]
Mode (Manual): [85], Count: 2

Mean, Mode and Median Visualization

# import libraries
import numpy as np
import pandas as pd
from collections import Counter
import matplotlib.pyplot as plt
# define your data and compute statistics here
data = [85, 90, 78, 92, 88, 91, 85, 87, 89, 86]
mean_np   = np.mean(data)
median_np = np.median(data)
# for mode, use Counter
counter   = Counter(data)
max_count = max(counter.values())
modes     = [k for k,v in counter.items() if v==max_count]
mode_val  = modes[0]

Distribution of Test Scores with Central Tendency Measures

Real-world Applications

5 minutes

Business Applications

  • Customer Satisfaction: Mean rating shows overall satisfaction, median shows typical experience
  • Sales Data: Mode identifies best-selling products, median shows typical sale amount
  • Employee Performance: Mean for overall team performance, median for typical employee

Educational Applications

  • Test Scores: Mean for class average, median for typical student performance
  • Grade Distribution: Mode shows most common grade
  • Attendance: Mean for overall attendance rate

Healthcare Applications

  • Patient Wait Times: Median often preferred due to skewed distributions
  • Treatment Outcomes: Mean for overall effectiveness, mode for most common result
  • Vital Signs: All three measures provide different insights

Interpretation Guidelines

  1. Always consider the context of your data
  2. Report multiple measures when appropriate
  3. Be aware of data distribution shape
  4. Consider the presence of outliers
  5. Choose the most appropriate measure for your specific question

Key Takeaways

Remember These Points

  • Data type determines which statistics are appropriate
  • Mean uses all data but is sensitive to outliers
  • Median is robust and represents the middle value
  • Mode identifies the most common value and works with all data types
  • Context matters when choosing and interpreting measures
  • Python provides powerful tools for calculating descriptive statistics

Next Lecture Preview

Descriptive Statistics Part II will cover:

  • Measures of variability (range, variance, standard deviation)
  • Measures of position (percentiles, quartiles, z-scores)
  • Shape of distributions (skewness and kurtosis)
  • Advanced Python visualization techniques

Practice Problems

Try These Problems

  1. Calculate mean, median, and mode for: 12, 15, 18, 12, 20, 25, 12, 30

  2. A dataset has mean = 50 and median = 45. What does this tell you about the distribution?

  3. Why might median be preferred over mean for reporting household income?

  4. Create a Python function to identify the most appropriate measure of central tendency for a given dataset.

Questions?

Thank you for your attention!

References & Further Reading

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