Descriptive Statistics

Statistics for Psychology: Descriptive Statistics

The Story of Understanding Numbers 📊

Imagine you’re a detective, and instead of solving crimes, you’re solving the mystery of what people think, feel, and do. But here’s the thing—you can’t interview everyone in the world! So you collect clues (data) and use special tools to understand what’s really going on.

These tools? They’re called Descriptive Statistics. Think of them as your detective’s magnifying glass, helping you see patterns that are invisible to the naked eye.

What Are Descriptive Statistics?

Descriptive statistics are like taking a photograph of your data. They summarize and describe what you’ve collected so you can understand it at a glance.

The Classroom Example 🎒

Imagine you asked 10 students: “How many hours did you study last week?”

Their answers: 2, 3, 3, 4, 5, 5, 5, 6, 7, 10

Just looking at these numbers is confusing! But with descriptive statistics, we can answer:

• What’s the typical amount of studying? (Central tendency)
• How different are students from each other? (Variability)
• Is the data balanced or lopsided? (Distribution)

Measures of Central Tendency

Central tendency tells you the “center” of your data—the typical or average value.

Think of it like asking: “If I had to describe this whole group with just ONE number, what would it be?”

The Three Musketeers of Central Tendency

1. The Mean (Average) 🎯

The mean is what most people call the “average.”

How to find it: Add all numbers together, then divide by how many numbers you have.

Example: Study hours: 2, 3, 3, 4, 5, 5, 5, 6, 7, 10

• Add them: 2+3+3+4+5+5+5+6+7+10 = 50
• Count them: 10 students
• Mean = 50 ÷ 10 = 5 hours

Real-world use: A psychologist studying stress levels might find the mean stress score of college students is 6.2 out of 10.

2. The Median (Middle Value) 🎪

The median is the middle number when you line everything up in order.

How to find it: Arrange numbers from smallest to largest. The middle one is your median!

Example: Study hours in order: 2, 3, 3, 4, 5, 5, 5, 6, 7, 10

With 10 numbers (even count), take the middle two (5 and 5) and average them:

• Median = (5 + 5) ÷ 2 = 5 hours

Why use median? It’s not fooled by extremes!

Imagine one student studied 50 hours (instead of 10):

• New mean = 94 ÷ 10 = 9.4 hours (way off!)
• New median = still 5 hours (accurate!)

3. The Mode (Most Common) 🏆

The mode is simply the number that appears most often.

Example: Study hours: 2, 3, 3, 4, 5, 5, 5, 6, 7, 10

The mode is 5 because it appears 3 times—more than any other number!

Fun fact: You can have:

• No mode (all values appear once)
• One mode (unimodal)
• Two modes (bimodal)
• Many modes (multimodal)

When to Use Which?

Measures of Variability

Central tendency tells you the center—but that’s only half the story!

Variability tells you how spread out or clustered your data is.

The Ice Cream Shop Analogy 🍦

Two ice cream shops both sell an average of 50 cones per day.

• Shop A: Sells 48, 49, 50, 51, 52 cones (very consistent!)
• Shop B: Sells 10, 30, 50, 70, 90 cones (wildly unpredictable!)

Same mean, VERY different stories. That’s why variability matters!

The Key Measures of Spread

1. Range 📏

The range is the simplest measure—just the difference between the highest and lowest values.

Formula: Range = Maximum - Minimum

Example: Study hours: 2, 3, 3, 4, 5, 5, 5, 6, 7, 10

• Range = 10 - 2 = 8 hours

Limitation: One extreme score can make the range misleading!

2. Variance 📐

Variance tells you, on average, how far each score is from the mean (squared).

Steps:

1. Find the mean
2. Subtract the mean from each score (these are “deviations”)
3. Square each deviation
4. Average the squared deviations

Example (simplified): Scores: 2, 4, 6 (Mean = 4)

• Deviations: -2, 0, +2
• Squared: 4, 0, 4
• Variance = (4+0+4) ÷ 3 = 2.67

3. Standard Deviation (SD) ⭐

The standard deviation is the square root of variance. It tells you, in the original units, how much scores typically differ from the mean.

Example: If variance = 2.67, then SD = √2.67 ≈ 1.63

Why we love SD:

• It’s in the same units as your data
• Easy to interpret: “On average, students study about 1.6 hours more or less than the mean”

Quick Reference

The Normal Distribution

Now for something magical—the normal distribution, also called the bell curve! 🔔

Why Is It Called “Normal”?

When you measure almost anything about people—height, weight, IQ, reaction time—the data often forms this beautiful, symmetrical shape.

The Bell Curve’s Superpowers

graph TD
    A["Normal Distribution"] --> B["Symmetrical"]
    A --> C["Mean = Median = Mode"]
    A --> D["68-95-99.7 Rule"]
    B --> E["Left mirrors Right"]
    C --> F["All at the center"]
    D --> G["Predictable spread"]

Characteristics of the Normal Curve

1. Symmetrical: The left side is a mirror image of the right
2. Bell-shaped: Highest in the middle, tapering at both ends
3. Mean = Median = Mode: They’re all at the exact center
4. Defined by 2 things: The mean (center) and SD (width)

The 68-95-99.7 Rule 🎯

This is your secret weapon for understanding any normal distribution!

• 68% of scores fall within 1 SD of the mean
• 95% of scores fall within 2 SDs of the mean
• 99.7% of scores fall within 3 SDs of the mean

Example: IQ Scores

• Mean = 100, SD = 15

This means only about 2.5% of people have an IQ above 130!

Z-Scores and Percentiles

Z-Scores: Your Universal Translator 🌐

Different tests use different scales. How do you compare apples to oranges?

Z-scores convert ANY score into a standard language!

Formula:

Z = (Your Score - Mean) ÷ Standard Deviation

What Z-scores tell you:

• Z = 0: You’re exactly at the mean
• Z = +1: You’re 1 SD above the mean
• Z = -1: You’re 1 SD below the mean
• Z = +2: You’re 2 SDs above the mean (top ~2.5%!)

Z-Score Example 🎓

Situation: Sarah scored 75 on a psychology test.

• Class mean = 70
• SD = 5

Calculation: Z = (75 - 70) ÷ 5 = +1.0

Interpretation: Sarah scored 1 standard deviation above average—better than about 84% of the class!

Percentiles: Your Ranking 🏅

A percentile tells you what percentage of scores fall below yours.

• 50th percentile = Median (half below, half above)
• 90th percentile = Better than 90% of people
• 25th percentile = Better than only 25%

Z-Scores to Percentiles (Common Conversions)

Real-World Example: Comparing Different Tests

Problem: Who did better?

• Alex: 85 on Test A (Mean=80, SD=10)
• Jordan: 92 on Test B (Mean=85, SD=5)

Solution using Z-scores:

• Alex: Z = (85-80)÷10 = +0.5
• Jordan: Z = (92-85)÷5 = +1.4

Winner: Jordan! Even though 92 seems close to 85, Jordan performed relatively better compared to their class.

Putting It All Together 🧩

graph TD
    A["Raw Data"] --> B["Descriptive Statistics"]
    B --> C["Central Tendency"]
    B --> D["Variability"]
    B --> E["Distribution Shape"]
    C --> F["Mean, Median, Mode"]
    D --> G["Range, Variance, SD"]
    E --> H["Normal Curve"]
    H --> I["Z-scores"]
    I --> J["Percentiles"]

The Full Detective Toolkit

1. Collect data from your sample
2. Describe the center using mean, median, or mode
3. Describe the spread using range, variance, or SD
4. Check the shape—is it normal?
5. Convert to Z-scores to compare across different scales
6. Find percentiles to understand relative standing

Why This Matters for Psychology 🧠

Every time psychologists study:

• Depression symptoms
• Personality traits
• Memory performance
• Treatment effectiveness

…they use these exact tools!

You’re not just learning statistics—you’re learning the language of psychological science.

Key Takeaways 🎯

• Descriptive statistics summarize data to tell its story
• Mean is the average; median is the middle; mode is the most common
• Variability (SD, variance, range) shows how spread out data is
• The normal distribution is symmetrical and follows the 68-95-99.7 rule
• Z-scores standardize any score for comparison
• Percentiles show your rank among all scores

Now you’re ready to describe any dataset like a pro! 🌟