Data analysis: descriptive statistics and graphs

Introduction to Data Analysis

Once a psychologist has finished their research—whether it's an experiment, an observation, or a questionnaire—they are left with a pile of "raw data." This data is usually just a long list of numbers or words. To make sense of it all and see what it actually means, we use data analysis. In this chapter, you will learn how to turn those messy results into clear summaries using descriptive statistics and graphs.

1. Types of Data

Before we start calculating, we need to know what kind of data we have. Psychologists generally work with two main types of data:

Quantitative vs. Qualitative Data

• Quantitative Data: This is data in the form of numbers. It tells us "how much" or "how many." Example: The score someone got on a memory test or the time it took to complete a task.
• Qualitative Data: This is data in the form of words or descriptions. It tells us about the "quality" or meaning of something. Example: A participant describing their feelings during an interview.

Primary vs. Secondary Data

• Primary Data: This is data collected first-hand by the researcher specifically for their own study. Example: You conduct your own experiment in class and record the results.
• Secondary Data: This is data that already exists, collected by someone else for a different purpose. Example: A psychologist using government statistics or looking at the results of a study published 10 years ago.

Quick Review: Remember Quantitative = Quantity (numbers), and Qualitative = Quality (words).

2. Descriptive Statistics: Measures of Central Tendency

Measures of central tendency are "averages." They give us a single value that represents the "middle" of a set of data.

The Mean

The mean is the arithmetic average. You add up all the scores and divide by the total number of scores.

\( \text{Mean} = \frac{\text{Sum of all scores}}{\text{Number of scores}} \)

Example: If scores are \( 5, 7, 8, \text{ and } 10 \), the mean is \( (5+7+8+10) \div 4 = 7.5 \).

The Median

The median is the middle score in a list of numbers when they are put in order from smallest to largest.

Example: For scores \( 2, 5, 8, 10, 12 \), the median is \( 8 \). If there are two middle numbers, the median is the average of those two.

The Mode

The mode is the score that appears most often in a data set.

Example: In the set \( 3, 5, 5, 6, 8, 9 \), the mode is \( 5 \).

Summary:
- Mean: The "maths" average (uses all data points).
- Median: The "middle" value.
- Mode: The "most" frequent value.

3. Measures of Dispersion: The Range

While central tendency tells us about the middle, dispersion tells us how "spread out" the scores are. In this course, we focus on the range.

To calculate the range, subtract the lowest score from the highest score. Usually, we add \( 1 \) to the result to account for rounding.

\( \text{Range} = (\text{Highest score} - \text{Lowest score}) + 1 \)

Example: If the highest score is \( 20 \) and the lowest is \( 5 \), the range is \( (20 - 5) + 1 = 16 \).

Why use it? A high range means the scores are very spread out. A low range means the participants scored very similarly.

4. Working with Numbers and Calculations

In the exam, you may be asked to perform basic calculations. Don't worry—you can use a calculator!

Percentages and Fractions

To find a percentage, divide the part by the whole and multiply by \( 100 \).

\( \text{Percentage} = \frac{\text{Score}}{\text{Total Possible}} \times 100 \)

Ratios: These compare two amounts. Example: If \( 10 \) people said "Yes" and \( 5 \) said "No," the ratio is \( 10:5 \), which simplifies to \( 2:1 \).

Decimals and Standard Form

Standard Form is used to write very large or very small numbers simply. It is written as \( a \times 10^n \).

Example: \( 50,000 \) in standard form is \( 5 \times 10^4 \).

Significant Figures and Estimation

• Significant Figures: You might be asked to round your answer. For example, rounding \( 7.567 \) to \( 2 \) significant figures gives \( 7.6 \).
• Estimation: Sometimes it’s helpful to "guess-timate" an answer first to make sure your final calculation makes sense. Round numbers to the nearest \( 10 \) or \( 100 \) to check.

5. Graphs and Tables

Psychologists use graphs to make data easy to see at a glance. You need to know how to construct (draw) and interpret (read) these:

Frequency Tables

A table used to record how often each "event" or "score" occurs. It often uses a tally system.

Bar Charts

Used for discrete data (data that fits into separate categories, like "Types of Phobias" or "Gender").
- The bars do not touch.
- The categories go on the x-axis (bottom) and the frequency goes on the y-axis (side).

Histograms

Used for continuous data (data that can be measured on a scale, like time or weight).
- The bars must touch because the data is continuous.
- The x-axis represents intervals (e.g., \( 0-5 \) seconds, \( 6-10 \) seconds).

Scatter Diagrams

Used to show the relationship (correlation) between two variables. Each dot represents one participant's score on both variables.

• Positive Correlation: As one variable goes up, the other goes up (dots go from bottom-left to top-right).
• Negative Correlation: As one variable goes up, the other goes down (dots go from top-left to bottom-right).
• Zero Correlation: There is no relationship; the dots are scattered randomly.

6. The Normal Distribution

If you measure something like height or IQ in a huge group of people, the results usually form a Normal Distribution curve (often called a "Bell Curve").

Key Characteristics:
- It is symmetrical (looks the same on both sides).
- The mean, median, and mode are all at the exact same point in the middle.
- Most people score near the average, with fewer people at the extreme high or low ends.

Did you know? Most human traits, like shoe size or brain weight, follow a normal distribution!

Key Takeaways for the Exam

1. Read the question carefully: Does it ask for the Mean, Median, or Mode?
2. Check your units: If a question asks for a percentage, make sure you include the \( \% \) sign!
3. Graphs: When drawing a bar chart, remember to leave gaps between the bars. When drawing a histogram, make sure the bars touch.
4. Labels: Always label your axes (e.g., "Number of Participants" or "Time in Seconds") and give your graph a title.