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Chapter 3: Problem 16

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

Expert verified
The distribution is right-skewed (positively skewed).

Step by step solution

01

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.
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

You recorded the time in seconds it took for 8 participants to solve a puzzle. These times appear below. However, when the data was entered into the statistical program, the score that was supposed to be 22.1 was entered as 21.2 . You had calculated the following measures of central tendency: the mean, the median, and the mean trimmed \(25 \%\). Which of these measures of central tendency will change when you correct the recording error? $$ \begin{array}{|c|} \hline \text { Time (seconds) } \\ \hline 15.2 \\ \hline 18.8 \\ \hline 19.3 \\ \hline 19.7 \\ \hline 20.2 \\ \hline 21.8 \\ \hline 22.1 \\ \hline 29.4 \\ \hline \end{array} $$

An experiment compared the ability of three groups of participants to remember briefly- presented chess positions. The data are shown below. The numbers represent the number of pieces correctly remembered from three chess positions. Compare the performance of each group. Consider spread as well as central tendency. $$ \begin{array}{|c|c|c|} \hline \text { Non-players } & \text { Beginners } & \text { Tournament players } \\ \hline 22.1 & 32.5 & 40.1 \\ \hline 22.3 & 37.1 & 45.6 \\ \hline 26.2 & 39.1 & 51.2 \\ \hline 29.6 & 40.5 & 56.4 \\ \hline 31.7 & 45.5 & 58.1 \\ \hline 33.5 & 51.3 & 71.1 \\ \hline 38.9 & 52.6 & 74.9 \\ \hline 39.7 & 55.7 & 75.9 \\ \hline 43.2 & 55.9 & 80.3 \\ \hline 43.2 & 57.7 & 85.3 \\ \hline \end{array} $$

Make up a dataset of 12 numbers with a positive skew. Use a statistical program to compute the skew. Is the mean larger than the median as it usually is for distributions with a positive skew? What is the value for skew?

You know the minimum, the maximum, and the \(25 \mathrm{th}, 50 \mathrm{th},\) and 75 th percentiles of a distribution. Which of the following measures of central tendency or variability can you determine? mean, median, mode, trimean, geometric mean, range, interquartile range, variance, standard deviation

True/False: When plotted on the same graph, a distribution with a mean of 50 and a standard deviation of 10 will look more spread out than will a distribution with a mean of 60 and a standard deviation of \(5 .\)
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