AP Stats Chapter 3 Quantitative Number Analysis
(D) When the extreme values are removed, the range, standard deviation, and variance will all decrease. The median, or middle value, remains the same if one extreme value at each end is removed, and it is possible that the mean remains unchanged.
A data set includes two outliers, one at each end. If both these outliers are removed, which of the following is a possible result? (A) Both the mean and standard deviation remain unchanged. (B) Both the median and standard deviation remain unchanged. (C) Both the standard deviation and variance remain unchanged. (D) Both the mean and median remain unchanged. (E) Both the mean and standard deviation increase.
(D) The interquartile range IQR = Q₃ - Q₁ = 13. The most commonly accepted definition of an outlier is any value greater than Q₃ + 1.5 (IQR) = 110.5 or less than Q₁ - 1.5 (IQR) = 58.5. Looking at the minimum and maximum values, we see that there are no outliers on the lower end, and at least one outlier on the upper end.
A random sample of golf scores gives the following summary statistics: n = 20, x̄ = 84.5 Sx = 11.5, minX = 68, Q₁ = 78, Med = 86, Q₃ = 91, maxX = 112. What can be said about the number of outliers? (A) 0 (B) 1 (C) 2 (D) At least 1 (E) At least 2
(A) All three have the same range: 22 - 2 = 20. The interquartile ranges of the first sets are 6, while the interquartile range of the middle set is 4. The medians are 12, 10, and 8, respectively. Only the lower distribution appears to be skewed right. With the given outliers, none of these distributions appears symmetric, and therefore, none appear normal (if the sample is small, any outliers in the sample call into question the use of t-procedures which depend upon a normality/large sample assumption).
Given these parallel boxplots, which of the following is true? (A) All three distributions have the same range. (B) All three distributions have the same interquartile range. (C) All three medians are between 9 and 13. (D) All three distributions appear to be skewed right. (E) All three distributions can reasonably be assumed to be of samples from normally distributed population.
(A) Note that adding 10 to every score in set A results in set B. Thus the means differ by 10, but measures of variability (for example, range, interquartile range, standard deviation, and variance) remain the same.
Given this back-to-back stemplot, which of the following is incorrect? (A) The distributions have the same mean. (B) The distributions have the same range. (C) The distributions have the same interquartile range. (D) The distributions have the same standard deviation. (E) The distributions have the same variance.
(E) Counting boxes (area) we note that Q₁ = 70 and Q₃ = 90. The interquartile range IQR is Q₃ - Q₁ = 90 - 70 = 20. Outliers would be values greater than Q₃ + 1.5 (IQR) = 120 or less than Q₁ - 1.5 (IQR) = 40. In this case, there are no such values.
Given this histogram, and using the most commonly accepted definition of outliers, what values would be considered outliers? (A) Between 115 and 120. (B) Between 110 and 120. (C) Between 50 and 55, or between 115 and 120. (D) Between 50 and 55, or between 110 and 120. (E) There are no outliers.
(B) Increasing every value by 20 percent will also increase Q₁, Q₃, and the IQR as well as both Q₁ - 1.5 (IQR) and Q₃ + 1.5 (IQR) by the same 20 percent. In other words, multiplying by a factor of 0.20 does not change how many outliers there are.
Using the most commonly accepted definition of outliers, a set has five outliers. If every value of the set in increased by 20 percent, how many outliers will there now be. (A) Fewer than five (B) Five (C) Six (D) More than six (E) It is impossible to determine without further information.
(E) The mean, standard deviation, variance, and range are all affected by outliers; the median and interquartile range are not.
When a set of data has suspect outliers, which of the following are preferred measures of central tendency and variability? (A) Mean and standard deviation (B) Mean and variance (C) Mean and range (D) Median and range (E) Median and interquartile range.
(C) Range is always reported a positive number. The sample COMES from the population, so the smallest value in the sample cannot be smaller than the smallest value in the population, and, similarly, the largest value in the sample cannot be larger than the largest value in the population. Outliers are extreme values, and while they affect the range, they do not affect the interquartile range which subtracts Q3 from Q1 (we say the range is sensitive to extreme values while the IQR is resistant to extreme values). The range is a single number, the largest value minus the smallest value, which is the same no matter how the set is arranged. IQR = Q₃ - Q₁ and the middle half of the data is between the quartiles Q₁ and Q₃.
Which of the following statements is incorrect? (A) The range of the sample data set can never be greater than the range of the population. (B) While the range is affected by outliers, the interquartile range is not. (C) Changing the order from ascending to descending changes the sign of the range. (D) The range is a single number, not an interval of values. (E) The interquartile range is the range of the middle half of the data.
