Introduction to Descriptive Statistics (List A)
Welcome to the world of Descriptive Statistics! After a psychologist finishes an experiment or a survey, they are often left with a huge pile of numbers called "raw data." On its own, raw data is hard to understand. Descriptive statistics are the tools we use to summarise and describe that data so we can see patterns and make sense of what happened in our research.
Think of it like a sports match: the raw data is every single pass and movement made during the game, but the descriptive statistics are the final score and the possession percentages. They give us the "big picture" quickly and clearly.
1. Measures of Central Tendency
These measures help us find the "typical" or "middle" score in a data set. There are three main types you need to know:
The Mean
This is the arithmetic average. You calculate it by adding up all the scores and dividing by the total number of scores (\(n\)).
Formula: \(\bar{x} = \frac{\sum x}{n}\)
- Strength: It is the most sensitive measure because it includes every single piece of data.
- Weakness: It can be easily distorted by anomalies (one or two extremely high or low scores).
The Median
This is the middle value when all the scores are placed in order from lowest to highest. If there is an even number of scores, the median is the average of the two middle numbers.
- Strength: It is not affected by extreme outliers/anomalies.
- Weakness: It doesn't take the actual value of all the scores into account, just their position.
The Mode
This is the most frequently occurring score in a data set.
- Strength: It is very easy to calculate and is the only measure that can be used for nominal data (data in categories, like "favourite colour").
- Weakness: A data set might have no mode, or it might have several (bimodal), which isn't very helpful.
Quick Review: To remember the difference, try this rhyme: "The Mean is the average; you add then divide. The Median is the middle; put the numbers in line. The Mode is the most; it's the one you see most of the time!"
2. Measures of Dispersion
While central tendency tells us where the middle is, dispersion tells us how spread out the scores are. Are they all bunched together, or are they widely scattered?
The Range
The range is the difference between the highest and lowest scores in a data set. Usually, in Psychology, we calculate it as: \(Highest\ Score - Lowest\ Score\).
- Strength: Very quick and easy to calculate.
- Weakness: Like the mean, it is easily affected by extreme scores.
Standard Deviation
This is a more sophisticated measure. it tells us the average distance of every score from the mean. A low standard deviation means the scores are clustered close to the mean; a high standard deviation means the scores are spread out.
The formula provided in your exam for the sample-estimate standard deviation is:
\(s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}\)
Don't worry if this formula looks scary! You don't need to memorise it, as it will be in your formula sheet. Just remember that it gives a much more accurate picture of the spread of data than the range does because it uses every score.
3. Percentages
Percentages are a way of expressing a number as a fraction of 100. They are excellent for comparing different groups of different sizes.
Calculation: \(\frac{Score}{Total} \times 100\)
Example: If 15 out of 60 participants obeyed the researcher, the percentage is \(\frac{15}{60} \times 100 = 25\%\).
4. Visualising Data: Graphs and Tables
Psychologists use several ways to present data visually so that the results are easy to "digest."
Frequency and Summary Tables
A frequency table shows how often each individual score or category occurs. A summary table shows the calculated descriptive statistics (like the mean and standard deviation) for each condition in an experiment.
Bar Charts
Used for discrete data (categories). The bars do not touch. For example, a bar chart might show the mean obedience scores for males vs. females.
Histograms
Used for continuous data (data on a scale). The bars must touch each other. This represents the fact that the data is on a continuous scale (like time or height).
Scatter Diagrams
These are used specifically for correlational research. Each dot represents one participant's score on two different variables (co-variables). (Cross-reference: See the "Correlational research" chapter for more on how to interpret these).
5. Distributions
When we plot data on a graph, the "shape" of the data tells us a lot about our participants.
Normal Distribution
This is the famous "Bell Curve." In a normal distribution, the mean, median, and mode are all at the same central point. Most people score in the middle, with very few people having extremely high or extremely low scores (e.g., IQ or height).
Skewed Distributions
Sometimes, data is not symmetrical. This is called a skew.
- Positive Skew: Most of the scores are at the lower end of the scale (the "hump" is on the left, and the long "tail" points to the right). Example: A very difficult exam where most people got low marks.
- Negative Skew: Most of the scores are at the higher end of the scale (the "hump" is on the right, and the long "tail" points to the left). Example: A very easy exam where most people got high marks.
6. Sense Checking
Sense checking is a vital skill for your exam. It simply means looking at your calculated answer and asking: "Does this actually make sense?"
Example: If you are calculating the mean score out of 20, and your answer comes out as 45.2, you have clearly made a mistake because the mean cannot be higher than the maximum possible score! Always take a second to look back at the raw data to see if your answer looks plausible.
Key Takeaways
- Central Tendency: Mean (average), Median (middle), Mode (most common).
- Dispersion: Range (difference) and Standard Deviation (spread from the mean).
- Graphs: Use Bar Charts for categories and Histograms for continuous scales.
- Distributions: Normal is a symmetrical bell; Skewed is lopsided.
- Accuracy: Always sense check your calculations!