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Chapter 4: Problem 13

In what sense is the variance (a) a type of mean? (b) not a readily understood measure of variability? (c) a stepping stone to the standard deviation?

Short Answer

Expert verified
Variance is a 'mean of squared differences' rather than a standard mean. Variance may not be easily interpretable as a measure of variability due to the square unit, making interpretation less intuitive. Finally, variance acts as a base for calculating standard deviation, as standard deviation squares the variance to return to the same unit, thereby making it a more comprehensible measure of dispersion.

Step by step solution

01

Mean of Variance

Variance is a statistical measurement of the dispersion of random variables. Rather than a 'type' of mean, it's more accurately described as a 'mean of squared deviations.' Here's why: Variance involves two steps: 1. Determine each data point's deviation from the mean (subtract the mean from each data point) 2. Square these deviations. The variance is then the mean of these squared deviations. We are taking a mean in the second step, but this mean is unique in that it involves squared values instead of raw data points.
02

Variance as a Measure of Variability

Variance poses interpretability problems because it involves squared units. For example, if we are measuring variance of heights (measured in inches), variance would be presented in square inches, which does not make much intuitive sense. Hence, Variance, although a precise statistical measurement, is not easily interpreted or understood directly, primarily because it's in squared units.
03

Variance and Standard Deviation

Standard deviation is simply the square root of variance. Taking the square root allows us to return to the original units of measure, thereby making our dispersion metric easily interpretable. Therefore, the variance computation is a necessary step to reach standard deviation. In calculating standard deviation, we square root the variance to return back to same unit of original data. This makes Standard Deviation more understandable and interpretable as a measure of dispersion.

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Key Concepts

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

Mean of Squared Deviations
Variance is often called the 'mean of squared deviations' because of the process used to calculate it. Let's break it down into simple steps:
  • First, find the mean (average) of your dataset.
  • Next, calculate how far each data point is from this mean, which is called the deviation.
  • Then, square these deviations. Squaring removes any negative signs, which helps in understanding the total spread.
  • Finally, compute the average of these squared values. This is the variance.
The reason variance is a 'mean' is because it's essentially an average of these squared deviations. But instead of averaging raw data, you average their squared differences. This particular characteristic means that variance captures the spread of data points around the mean effectively.
Measure of Variability
Variance serves as a measure of variability, giving us insight into how much the data points differ from the mean. However, it can be tricky to interpret because:
  • It involves squared units, which can be hard to intuitively grasp. For example, if the data is in meters, variance will be in square meters.
  • This squaring amplifies the differences, making variance appear larger compared to other measures, like standard deviation.
Despite these challenges, understanding variance is crucial because it is a fundamental concept in statistics that helps quantify variability. It informs us about the degree of spread in the dataset, although the '% of variance' isn't immediately usable without further processing.
Standard Deviation
The standard deviation is closely related to variance, acting as a more interpretable statistic. Here's how it connects:
  • Start with the variance, which is the squared measure of data spread.
  • Take the square root of the variance. This action brings the measurement back to the original units of the data, making it easier to understand.
  • As a result, standard deviation provides a practical understanding of data dispersion.
By converting variance into standard deviation, we make the data's variability understandable in the context of the original data units, such as meters instead of square meters. This makes it a preferred choice for data analysis, offering clear insights regarding dataset variability.

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Most popular questions from this chapter

Using the definition formula for the sum of squares, calculate the sample standard deviation for the following four scores: \(1,3,4,4 .\)

Add 10 to only the smallest score in Question 4.3(1,3,4,4) to produce another new distribution (11,3,4,4) . Would you expect the value of \(s\) to be the same for both the original and new distributions? Explain your answer, and then calculate s for the new distribution.

Assume that the distribution of IQ scores for all college students has a mean of \(120,\) with a standard deviation of \(15 .\) These two bits of information imply which of the following? (a) All students have an IQ of either 105 or 135 because everybody in the distribution is either one standard deviation above or below the mean. True or false? (b) All students score between 105 and 135 because everybody is within one standard deviation on either side of the mean. True or false? (c) On the average, students deviate approximately 15 points on either side of the mean. True or false? (d) Some students deviate more than one standard deviation above or below the mean. True or false? (e) All students deviate more than one standard deviation above or below the mean. True or false? (f) Scott's IQ score of 150 deviates two standard deviations above the mean. True or false?

Why can't the value of the standard deviation ever be negative?

Employees of Corporation A earn annual salaries described by a mean of \(\$ 90,000\) and a standard deviation of \(\$ 10,000\). (a) The majority of all salaries fall between what two values? (b) A small minority of all salaries are less than what value? (c) A small minority of all salaries are more than what value? (d) Answer parts (a), (b), and (c) for Corporation B's employees, who earn annual salaries described by a mean of \(\$ 90,000\) and a standard deviation of \(\$ 2,000\).
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