Question

If the mean time to respond to a stimulus is much higher than the median time to respond, what can you say about the shape of the distribution of response times?

Short answer

The distribution is right-skewed (positively skewed).

Step-by-step solution

Understanding the Terms

The problem refers to the mean and median of response times. The mean is the average of all response times, while the median is the middle value when response times are arranged in order. These measures help us understand the distribution of a data set.

Analyzing the Relationship

If the mean is much higher than the median, it indicates that there is a skewness in the data. Specifically, this situation often occurs when there are outliers (extremely high values) dragging the mean upwards.

Determining the Distribution Shape

When the mean is greater than the median, the distribution is typically right-skewed or positively skewed. This means that the majority of the data values are concentrated on the left, with a long tail stretching out towards the right.

Key concepts

Mean vs Median

When discussing data distributions, two key metrics often come up: the mean and the median. Both are measures of central tendency, but they can tell us different stories about a dataset.

• Mean: The mean, often called the average, is calculated by summing all the data values and dividing by the number of values. It can be influenced heavily by extreme values or outliers in the dataset.
• Median: The median is the middle value when all the data points are ordered from smallest to largest. If there is an even number of data points, it is the average of the two middle numbers. Unlike the mean, the median is robust against outliers and tends to provide a better measure of central tendency in skewed distributions.

By comparing the mean and median, we can begin to understand the shape and skewness of a data distribution.

Right-Skewed Distribution

A right-skewed distribution is a type of data distribution where the tail on the right side of the distribution is longer or fatter than the left side. This shape indicates that there are more low values and fewer high values. Here are some key features:

• Characteristic Shape: The graph of a right-skewed distribution slopes downward more gently on the right, leading to a tail on that side. This often happens in datasets where a small number of high outliers lift the mean above the median.
• Mean vs. Median: In right-skewed distributions, the mean is typically greater than the median. This is because the mean is influenced more by the long tail of high values.

Such a distribution is commonly seen in datasets like income levels, where a few extremely high incomes stretch the distribution to the right.

Outliers in Data

Outliers are data points that differ significantly from other observations in a dataset. They can occur naturally, such as extremely rare events or measurement errors. Their presence can affect data analysis significantly:

• Impact on Mean: Outliers can skew the mean, pulling it toward the extreme values, which may not accurately represent the data's central tendency.
• Minimal Effect on Median: The median remains stable in the presence of outliers, making it a more reliable measure of central tendency in skewed datasets.
• Identifying Outliers: Statistical methods, such as the interquartile range (IQR) or z-scores, can be used to identify and possibly eliminate outliers, depending on the context and the nature of the dataset.

Understanding outliers and how they influence statistical measures is an essential step in accurately interpreting data.