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Chapter 1: Q76SE (page 49)

The three measures of center introduced in this chapter are the mean, median, and trimmed mean. Two additional measures of center that are occasionally used are the midrange,which is the average of the smallest and largest observations, and the midfourth,which is the average of the two fourths. Which of these five measures of center are resistant to the effects of outliers and which are not? Explain your reasoning.

Short Answer

Expert verified

Mean and midrange is not resistant to the outliers.

Median, trimmed mean and mid-fourth are resistant to the outliers.

Step by step solution

01

Given information

The three measures of center are mean, median, and trimmed mean.

The two additional measures of the center that are used are midrange and mid-fourth.

02

Explain which measure of center is resistant to the effects of outliers

From the provided measures of center, mean, and midrange are computed by using the extreme values. This implies that they are not resistant to the outliers.

Median, trimmed mean and mid-fourth takes the position of values into account and the values themselves. This implies that they are most resistant to the outliers.

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Most popular questions from this chapter

Using a long rod that has length \({\rm{\mu }}\)you are going to lay out a square plot in which the length of each side is \({\rm{\mu }}\).Thus the area of the plot will be \({{\rm{\mu }}^{\rm{2}}}\)However, you do not know the value of \({\rm{\mu }}\), so you decide to make \({\rm{n}}\)independent measurements \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}\)of the length. Assume that each \({{\rm{X}}_{\rm{i}}}\)has mean \({\rm{\mu }}\) (unbiased measurements) and variance \({{\rm{\sigma }}^{\rm{2}}}\).

a. Show that \({{\rm{\bar X}}^{\rm{2}}}\)is not an unbiased estimator for \({{\rm{\mu }}^{\rm{2}}}\). (Hint: For any \({\rm{Y,E}}\left( {{{\rm{Y}}^{\rm{2}}}} \right){\rm{ = V(Y) + (E(Y)}}{{\rm{)}}^{\rm{2}}}\)Apply this with \({\rm{Y = \bar X}}\))

b. For what value of \({\rm{k}}\)is the estimator \({{\rm{\bar X}}^{\rm{2}}}{\rm{ - k}}{{\rm{S}}^{\rm{2}}}\)unbiased for \({{\rm{\mu }}^{\rm{2}}}\)? (Hint: Compute \({\rm{E}}\left( {{{{\rm{\bar X}}}^{\rm{2}}}{\rm{ - k\;}}{{\rm{S}}^{\rm{2}}}} \right)\))

Once an individual has been infected with a certain disease, let \({\rm{X}}\) represent the time (days) that elapses before the individual becomes infectious. The article proposes a Weibull distribution with \({\rm{\alpha = 2}}{\rm{.2}}\), \({\rm{\beta = 1}}{\rm{.1}}\), and \({\rm{\gamma = 0}}{\rm{.5}}\).

a. Calculate \({\rm{P(1 < X < 2)}}\).

b. Calculate \({\rm{P(X > 1}}{\rm{.5)}}\).

c. What is the \({\rm{90th}}\) percentile of the distribution?

d. What are the mean and standard deviation of \({\rm{X}}\)?

For each of the following hypothetical populations, give

a plausible sample of size 4:

a. All distances that might result when you throw a football

b. Page lengths of books published 5 years from now

c. All possible earthquake-strength measurements (Richter scale) that might be recorded in California during the next year

d. All possible yields (in grams) from a certain chemical reaction carried out in a laboratory.

Blood pressure values are often reported to the nearest5 mmHg (100, 105, 110, etc.). Suppose the actual bloodpressure values for nine randomly selected individuals are

118.6 127.4 138.4 130.0 113.7 122.0 108.3131.5 133.2

  1. What is the median of the reportedblood pressure values?
  2. Suppose the blood pressure of the second individual is 127.6 rather than 127.4 (a small change in a single value). How does this affect the median of the reported values? What does this say about the sensitivity of the median to rounding or grouping in the data?

The accompanying frequency distribution on depositedenergy (mJ) was extracted from the article 鈥淓xperimentalAnalysis of Laser-Induced Spark Ignition of LeanTurbulent Premixed Flames鈥� (Combustion and Flame,2013: 1414鈥�1427).

1.0鈭�< 2.0 5 2.0鈭�<2.4 11

2.4鈭�< 2.6 13 2.6鈭�<2.8 30

2.8鈭�< 3.0 46 3.0鈭�< 3.2 66

3.2鈭�<3.4 133 3.4鈭�< 3.6 141

3.6鈭�< 3.8 126 3.8鈭�< 4.0 92

4.0鈭�< 4.2 73 4.2鈭�< 4.4 38

4.4鈭�< 4.6 19 4.6鈭�< 5.0 11

a. What proportion of these ignition trials resulted in adeposited energy of less than 3 mJ?

b. What proportion of these ignition trials resulted in adeposited energy of at least 4 mJ?

c. Roughly what proportion of the trials resulted in adeposited energy of at least 3.5 mJ?

d. Construct a histogram and comment on its shape.
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