A college statistics class conducted a survey of how students spend their money. They asked 25 students to estimate how much money they typically spend each week on fast food. They determined that the mean amount spent on fast food is $31.52 with a standard deviation of $21.60.
Later they realized that a value entered as $3 should have been $30. They recalculate the mean and standard deviation. The mean is now $32.60.
Which of the following is true about the standard deviation?
a. The standard deviation will increase, because we have increased the value of a data point.
b.The standard deviation will stay the same, because the standard deviation is not affected by a change in a single measurement.
c.The standard deviation will decrease, because this change moved a data point closer to the mean.
c. The standard deviation will decrease, because this change moved a data point closer to the mean.
The correct answer is c. The standard deviation will decrease because this change moved a data point closer to the mean.
To understand why the standard deviation will decrease, it is important to understand what the standard deviation represents. The standard deviation measures the amount of variation or dispersion in a set of data points. It tells us how spread out the data points are from the mean.
When a value that is far from the mean is corrected or moved closer to the mean, it reduces the overall variability or spread of the data. In this case, the incorrect value of $3 was replaced with the corrected value of $30, which is much closer to the mean of $31.52. As a result, the distance between the individual data point and the mean has decreased, leading to decreased variability and a smaller standard deviation.
Remember, the standard deviation is sensitive to the distance between individual data points and the mean. Therefore, correcting a data point that is far from the mean will generally result in a decrease in the standard deviation.