Chapter 3 STATS

mean note: The mean is another term for the arithmetic average. It can be thought of as the balancing point of a distribution of data.

The mean of a collection of data is located at the​ "balancing point" of a distribution of data.

note: The symbol ∑ stands for summation. If x represented a single​ observation, then ∑x would mean that all the values should be added together.

A. For distributions that are roughly symmetric

answer: standard deviation note: For most​ distributions, a majority of observations are within one standard deviation of the mean value.

D. Within one standard deviation of the mean

note: The variance is the square of the standard deviation. It is represented symbolically by s squared s2.

A. The units for the variance are always squared. note: The units for the variance are always squared. Measuring spread distance with the variance implies that the units for measuring spread are different from the units for measuring​ center, which is not true.

A. The center of the distribution of salaries for men in the country is greater than the center for women. note: In a distribution of​ values, the typical value is given by the mean. In this​ case, the average salary of a man is higher than that of a​ woman, so when comparing the distribution of​ men's salaries to​ women's salaries, the center of the distribution for men is greater than the center of the distribution for the women.

note: Since the difference between running times in the​ 100-meter event will be within a few seconds of each​ other, the running times will have small variation. In the​ marathon, since the running times are likely to be minutes​ apart, the times will have greater variation.​ Thus, the marathon running times will have a greater standard deviation.

note: According to the Empirical​ Rule, if a distribution is unimodal and​ symmetric, approximately​ 95% of the observations will be within two standards deviation of the mean.

answer: symmetric and unimodal. note: According to the Empirical​ Rule, if a distribution is unimodal and​ symmetric, approximately​ 68% of the observations will be within one standard deviation of the​ mean, approximately​ 95% of the observations will be within two standard deviations of the​ mean, and nearly all the observations will be within three standard deviations of the mean.

A. How many standard deviations away an observation is from the mean note: A standard unit is how many standard deviations away an observation is from the mean. A measurement converted to standard units is called a​ z-score.

D. The observation is equal to the mean. note: If an observation has a​ z-score of​ 0, then it is equal to the mean. The mean is 0 standard deviations away from​ itself, so it has a​ z-score of 0.

answer: ​z-score note: The​ z-score measures distance from a mean in terms of standard​ deviations, so it can be used to compare values measured in different​ units, such as inches and pounds.

median note: The median is the value that would be right in the middle if you were to sort the data from smallest to largest. About​ 50% of the observations are below it and about​ 50% of the observations are above it.

C. To give a typical value of a data set note: The median is a typical value of a data set. It is used particularly when the distribution is skewed.

Skewed note: The median is often used for skewed distributions. The mean is not often used for skewed distributions because skew affects the mean more than it affects the median.

note: The mean and standard deviation are used to measure the center and​ variation, respectively, when a distribution is symmetric.

note: The interquartile range tells us how much space the middle​ 50% of the data occupy. It is found by subtracting the third quartile from the first quartile.

median and mean

B. The median is preferred when the data is strongly skewed or has outliers. note: The median provides a better measure of center when the data is skewed or has outliers because the presence of an outlier has a much greater effect on the mean.

A. The mean is preferred when the data is relatively symmetric.

B. The mean tends to be greater than the median. note: The mean tends to be greater than the median in a​ right-skewed distribution. This is because the higher values to the right of the center pull the mean up more than they affect the median.

note: If the mean and the median of a distribution are approximately the​ same, then the shape of the distribution is likely to be symmetric.

note: The median is resistant to​ outliers, so when a distribution contains​ outliers, the median is the best choice for a measure of center.

note: The median is resistant to outliers. This makes it a good choice for a measure of center when a distribution is skewed.

D. The medians and interquartile ranges note: When comparing two​ distributions, one should always use the same measures of center and spread for both distributions. Since the mean will be affected by the skew or outliers in the first​ distribution, use the median and interquartile range for both distributions.

Outliers are observed values far from the main group of data. In a histogram they are separated from the others by space. Outliers must be looked at in closer context to know how to treat them. If they are​ mistakes, they might be removed or corrected. If they are not​ mistakes, you might do the analysis​ twice, once with and once without the outliers. note: Outliers are observed values that lie outside the range of the main group of data. When an outlier is​ present, the observer needs to consider it more closely. Sometimes it is just a mistake that happened while collecting the data and can be corrected or discarded. Other times it is a legitimately observed value. In those​ cases, the analysis needs to be presented once with outliers and once without outliers to give a better idea of what a typical value is. Note that in​ statistics, potential outliers are defined as observations that are more than 1.5 interquartile ranges below the first quartile or above the third​ quartile, not above or below the median. Also note that a potential outlier is not the same thing as an outlier.

The median is more​ resistant, which indicates that it usually changes less than the mean when comparing data with and without outliers. note: The median is more resistant to outliers than the​ mean, especially when the outliers have extreme values. The presence of an extreme value can cause the mean to become very skewed because it will shift heavily in the direction of the extreme value. The amount the median shifts by is based on the number of data​ observations, because it is determined by the middle value after ordering all the observations from lowest to highest. If there is only one outlier with an extremely large value the median will shift very​ slightly, while the mean will change significantly.

note: The median is resistant to outliers and extreme values because it orders the data from lowest to highest and looks at the middle value. The highest value does not change the​ order, and so it does not change the median. The mean is the balancing point for the data set. When looking at the shape of a​ histogram, the mean is the point which balances the weight on both sides. If an extreme value is placed on one end of the​ mean, it has to shift in that direction to keep everything balanced.

note: The median gives a better measure of center for this distribution because the​ professor's age is an outlying observation. The median also tends to give a better representation of a typical observation in a skewed distribution.

median.

note: The length of the box in a boxplot is proportional to the IQR. The left edge of the box is at the first quartile and the right edge is at the third quartile.

In a​ boxplot, potential outliers are points that are more than 1.5 IQRs below the first quartile or above the third quartile.

note: In a​ boxplot, the whiskers extend to the most extreme values that are not potential outliers. Potential outliers are then represented by others​ markers, such as dots.

The first step with potential outliers is always to investigate. A potential outlier might not be an outlier at all. Or a potential outlier might tell an interesting​ story, or it might be the result of an error in entering data.

Correct answer: Bimodal or any multimodal distribution Boxplots are best used only for unimodal distributions because they hide bimodality or any multimodality.

The mean is not shown in a​ boxplot, so it is not used to construct boxplots.