Problem 3 - An Introduction to Psychological Statistics

Question

Your younger brother comes home one day after taking a science test. He says that some- one at school told him that “60% of the students in the class scored above the median test grade.” What is wrong with this statement? What if he had said “60% of the students scored above the mean?”

Step-by-Step Solution

Answer

The median claim is incorrect; the mean claim may or may not be true.

Step 1: Understanding the Median

The median is the middle value of a data set when it is ordered from least to greatest. By definition, 50% of the observations fall below the median and 50% fall above it.

Step 2: Identifying the Error

If someone claims that "60% of the students scored above the median," this is incorrect because it violates the definition of the median. Only 50% of students should score above it.

Step 3: Understanding the Mean

The mean is the average of a data set, calculated by summing all the values and dividing by the number of observations. Unlike the median, the mean is affected by extreme values or skewed distributions.

Step 4: Analyzing the Second Statement

If someone claims that “60% of the students scored above the mean,” this could theoretically be true, depending on the distribution of the scores. In skewed distributions, a majority can be above or below the mean.

Key Concepts

Median in Statistics

When dealing with data, understanding the concept of the median is essential. The median is the middle value that separates the higher half from the lower half of a data set. Imagine you have a list of test scores arranged in numerical order from smallest to largest. The median is the number that falls right in the center.
Here's the important part:

• By its definition, half of the data points will be below this number, and half will be above it.
• Therefore, it is a measure of central tendency that is not swayed by extremely high or low values in the data set.

This is why the statement "60% of the students scored above the median" is incorrect. By definition, only 50% can be above it.

Mean in Statistics

The mean, often referred to as the average, is another measure of central tendency. It is calculated by adding together all the numbers in a data set and then dividing by the total number of values. While it gives us the average score, the mean is different from the median because

• It can be heavily influenced by outliers, which are unusually high or low values in the data set.
• It does not always split the dataset into two equal halves of frequency.

Therefore, in situations where data is skewed, more (or fewer) than 50% of data points might fall above the mean.

Distribution of Data

Distribution of data refers to the way values in a data set are spread or arranged. Picture a graph of test scores. The distribution tells us how scores are spread along the number line, revealing patterns that might not be immediately apparent.
Some typical patterns include:

• Symmetrical Distribution: When data is evenly distributed around a central value, often leading to the mean and median being similar.
• Asymmetrical or Skewed Distribution: Here, data is stretched more to one side, causing the mean and median to differ.

Understanding the distribution is crucial while interpreting mean and median, especially in datasets where the scores are not evenly distributed.

Skewed Distribution

In skewed distributions, the values at either end of the data set have a notable impact. When a distribution is skewed, it means that one tail of the data is longer or fatter than the other. There are two primary types of skewed distributions:

• Left-Skewed (Negative Skew): Most data points are high, and the tail points to the left.
• Right-Skewed (Positive Skew): Most data points are low, and the tail points to the right.

The mean is pulled in the direction of the skewness. So, in a right-skewed distribution, the mean is usually greater than the median, which could explain why more than 50% of students might score above it. Understanding skewness is essential in data analysis because it impacts the interpretation of results, particularly concerning the mean.