Log in. Study now. See Answer. Best Answer. Study guides. Statistics 20 cards. What are the brain's association areas. What is a field hockey stick made of. How old is she is rebecca stevenson. When during pregnancy should one quit smoking. Q: Can interquartile range be zero Write your answer Related questions. Is it possible for interquartile range to be zero? What is the interquartile range of ?
Why is the interquartile range a more appropriate measure for spread than the range? What is considered measures of variabilty is it mean interquartile range or mean absolute deviation? What is the meaing of interquartile range in math? Does a box plot represent an interquartile range? What does interquartile range mean? What is unaffected by outliers? What includes the range and the interquartile range? Does an outlier have a greater effect on the standard deviation or interquartile range?
What is the interquartile range a measure of? The interquartile range is used as a measure of variability to overcome what difficulty of the range? What is the IQ range?
What is the range between the upper quartile and lower quartile called? What is interquartile range? What is the semi interquartile range? Affine transformations have the equation of a line:. Some books call this a linear transformation , because it has the equation of a straight line. The measures of location and spread introduced in this chapter behave quite regularly when a list is transformed by an affine transformation.
Some of these relations are derived in a footnote. Using these relations can simplify calculating measures of location or spread when the units of measurement are changed.
The following exercise checks your ability to use these rules. Measures of location and spread can tell us a great deal about lists of numbers. For example, for any list, at least half the numbers in the list are no larger than the median , and at least half the numbers in the list are at least as large as the median this is one way of defining the median.
The mean and SD also can tell us about the fractions of values in a list in various ranges. What is the smallest the mean of the list could be? More generally, if any particular fraction of values in a list exceeds a given threshold, and none of the values in the list is negative, then the mean of the list cannot be arbitrarily small. Markov's inequality turns this idea upside down to limit the fraction of numbers in a list that can exceed any given threshold, provided the list contains no negative number.
The limit depends on the mean of the list, and the threshold: see. If the mean of a list of numbers is M , and the list contains no negative number then. There are students in a class. No student can have a negative amount of money in his or her pocket, so Markov's inequality applies. Markov's inequality guarantees that. If we know the mean of a list and its SD, we know something about how many of the numbers in the list must be in various ranges.
How small could the SD of the list be? Thus the SD of the list must be at least. More generally, if a particular fraction of the values differ from the mean of the list by at least a given threshold, then the SD of the list cannot be too small.
Chebychev's inequality turns this around to find a bound on the fraction of numbers in the list that differ from the mean by more than any given threshold. The bound depends on the SD of the list and the threshold. If the mean of a list of numbers is M and the standard deviation of the list is SD, then for every positive number k ,.
A heuristic derivation of Chebychev's inequality is in a note. Chebychev's inequality says that not too many of the numbers in a list can be far from the mean, where far is measured in standard deviations. Conversely, if a large fraction of the values are far from the mean, the SD of the list must be large. The mean weight of students in a certain class of students is lbs.
What fraction weighs between 90 lbs. We cannot get an exact answer, but we can get a lower bound using Chebychev's inequality. The range from 90 lbs. Thus the fraction who weigh between 90 lbs.
In some problems, it is possible to apply both Markov's inequality and Chebychev's inequality. When that happens, use whichever inequality gives the more precise answer—that is, the inequality that limits the fraction most stringently. On the average, it takes 45 minutes to cross the San Francisco Bay Bridge during rush hour.
The SD of the time it takes to cross the bridge is 15 minutes. What's the largest fraction of trips for which it could take more than 2 hours to cross the bridge?
Travel time is positive, so we can use Markov's inequality. By Markov's inequality,. This is not a possible travel time it always takes a positive amount of time to cross the bridge. By Chebychev's inequality,. This is a more restrictive bound than the one Markov's inequality gives in this problem Markov's inequality gave Larger lower bounds are better; smaller upper bounds are better. The following exercises check your ability to apply Markov's inequality and Chebychev's inequality.
This chapter introduced several ways to summarize lists of numbers, quantitative data. Some summaries, measures of location , seek to be as close as possible to every element of the list—to typify the elements. The mean , median , and mode are examples: They represent typical values of the list. The mean, median, and mode each are "as close as possible" to all the elements in the list, for different definitions of the proximity of two numbers: for the mean, the distance between two numbers is the square of their difference; for the median, the distance between two numbers is the absolute value of their difference; and for the mode, the distance between two numbers is 1 if the numbers differ, 0 if they are equal.
The mean is the sum of the elements, divided by the number of elements. The median is the smallest element that is at least as large as at least half the elements. The mode is the most common value in the list.
The mode makes sense for qualitative and categorical data as well as quantitative data, but the mean and median make sense only for quantitative data. The mean, median, and mode differ in their sensitivity to changes to the data, or resistance.
A statistic that can be changed arbitrarily by altering a single datum is not resistant. The median is resistant. The mean is not resistant. The resistance of the mode depends on the distribution of values in the list. The rms root mean square measures the average size of the elements of a list, without regard to their signs.
The rms is not resistant. Other summaries, measures of spread , reflect how the values of the list differ from each other. The range of a list of numbers is the largest number minus the smallest number.
The range is zero if and only if all the numbers in the list are equal. The range is not resistant. The SD measures the average size of the differences between the mean and the elements of the list: It is the rms of the list of deviations from the mean.
The SD of a list is zero if and only if all the numbers in the list are equal. The SD is not resistant. The IQR is the upper quartile minus the lower quartile. The IQR is resistant. If the units of measurement change by an affine transformation, measures of location and spread in the new units of measurement have simple relationships to their values in the old units. Measures of location and spread contain a surprising amount of information about lists of numbers: Markov's inequality limits the fraction of elements of the list that exceed any given threshold, in terms of the mean of the list and the threshold, provided the list contains no negative number.
Chebychev's inequality limits the fraction of elements whose difference from the mean of the list exceeds any given threshold, in terms of the SD of the list and the threshold. Measures of location Measures of location summarize a list of numbers by a "typical" value.
Measures of Spread Measures of spread summarize how much members of a list of numbers differ from each other. A heuristic derivation of Markov's inequality is in a note. Lucas Farias 1, 1 1 gold badge 8 8 silver badges 22 22 bronze badges. Then your six statistics would be: 0 0 0 1 0 1 following the order you presented yours. Add a comment. Active Oldest Votes. Improve this answer. Noah Noah To see this, start by making a data set with the IQR you want; for example you can get an IQR of 0 by making a lot of values just over half will do in the middle of your data set the same, like so: 1 2 3 5 5 5 5 5 5 5 7 8 9 13 values, 7 of them equal and in the middle; in this case the mean is 5 Now add any number you choose to all the values.
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