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Chapter 2: Problem 30

For each of the following variables, would you use the median or mean for describing the center of the distribution? Why? (Think about the likely shape of the distribution.) a. Amount of liquid in bottles of capacity one liter b. The salary of all the employees in a company c. Number of requests to reset passwords for individual email accounts.

Short Answer

Expert verified
a. Mean; b. Median; c. Median, based on each variable's distribution shape.

Step by step solution

01

Understanding the Distribution

Consider the distribution of each variable to determine the shape: a. The amount of liquid in bottles often follows a normal distribution as manufacturing processes typically ensure consistency, so slight variations will be symmetrical. b. Salaries in a company usually exhibit skewness because a small number of employees (like executives) earn substantially more than others, causing the distribution to be right-skewed. c. The number of password reset requests is expected to be right-skewed because most users request a reset infrequently, but a few may request frequently due to forgetfulness.
02

Choose Median or Mean Based on Distribution

Based on the distribution shape determined in Step 1: a. For the amount of liquid in bottles, use the **mean** because the normal distribution implies symmetry and the mean will accurately reflect the center. b. For the salary of all employees in a company, use the **median** because the right-skewed distribution makes the median a better measure to avoid the influence of outliers. c. For the number of password resets, use the **median** to best represent the average user, as the distribution is likely right-skewed.

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Key Concepts

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

Median
The median is a measure of central tendency that represents the middle value of a data set when it is ordered from smallest to largest. When dealing with skewed data, the median is often preferable to the mean because it is less influenced by outliers.
This makes it a robust measure that accurately reflects the center of a distribution that is not symmetrical.
  • In a right-skewed distribution, such as the salary data, the few high salaries can skew the mean upwards, but the median remains unaffected.
  • Similarly, in data like password reset requests, where most values are low and a few are high, the median gives a clearer picture of the typical value.
Mean
The mean is the arithmetic average of a set of numbers. It is calculated by adding all the numbers together and then dividing by the count of numbers. The mean is most useful when the data distribution is symmetrical because it takes into account all data points.
It gives a balance point of the data set and works well for normally distributed data without extreme outliers.
  • For example, in the case of the liquid in bottles, the production process often aims for consistency, leading to a symmetric distribution. Here, the mean provides an accurate measure of the central tendency.
  • In contrast, when there are outliers, the mean can be misleading, as these extreme values can "pull" the mean away from the center of the data.
Distribution Shape
The shape of a data distribution provides insights into how data points are spread out. It helps determine which measure of central tendency is most appropriate. Common distribution shapes include:
  • Normal (Symmetrical): Data is evenly distributed around the center, indicative of processes under precise control, like the liquid in bottles.
  • Right-Skewed (Positive Skew): Most data points are clustered at the lower end, with a long tail extending to the right, often found in salary distributions or password resets.
  • Left-Skewed (Negative Skew): Opposite of right-skewed, with most data at the higher end and tail extending to the left, although less common in natural occurrences.
Understanding the shape of the distribution aids in choosing whether the median or mean is appropriate.
Skewness
Skewness refers to the measure of the asymmetry of the probability distribution of a real-valued random variable. It indicates the direction and degree to which a distribution deviates from a normal distribution.
A skewed distribution can be either:
  • Right (Positive) Skew: Tail is longer on the right, common in datasets like salary or password reset requests where a few items lie far to the right of the majority.
  • Left (Negative) Skew: Tail is longer on the left, though this configuration is less frequent in typical business or manufacturing data.
In situations with significant skewness, especially when deciding between median and mean, the median often emerges as a better representative of the center as it isn't dragged by extreme values as the mean is. Understanding skewness helps to correctly interpret the data and use the appropriate statistical measure.

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Most popular questions from this chapter

A study of 13 children suffering from asthma (Clinical and Experimental Allergy, vol. \(20,\) pp. \(429-432,1990\) ) compared single inhaled doses of formoterol (F) and salbutamol (S). Each child was evaluated using both medications. The outcome measured was the child's peak expiratory flow (PEF) eight hours following treament. Is there a difference in the PEF level for the two medications? The data on PEF follow: $$ \begin{array}{ccc} \hline \text { Child } & \mathbf{F} & \mathbf{S} \\ \hline 1 & 310 & 270 \\ 2 & 385 & 370 \\ 3 & 400 & 310 \\ 4 & 310 & 260 \\ 5 & 410 & 380 \\ 6 & 370 & 300 \\ 7 & 410 & 390 \\ 8 & 320 & 290 \\ 9 & 330 & 365 \\ 10 & 250 & 210 \\ 11 & 380 & 350 \\ 12 & 340 & 260 \\ 13 & 220 & 90 \\ \hline \end{array} $$ a. Construct plots to compare formoterol and salbutamol. Write a short summary comparing the two distributions of the peak expiratory flow. b. Consider the distribution of differences between the PEF levels of the two medications. Find the 13 differences and construct and interpret a plot of the differences. If on the average there is no difference between the PEF level for the two brands, where would you expect the differences to be centered?

National Geographic Traveler magazine recently presented data on the annual number of vacation days averaged by residents of eight countries. They reported 42 days for Italy, 37 for France, 35 for Germany, 34 for Brazil, 28 for Britain, 26 for Canada, 25 for Japan, and 13 for the United States. a. Report the median. b. By finding the median of the four values below the median, report the first quartile. c. Find the third quartile. d. Interpret the values found in parts a-c in the context of these data.

France is most popular holiday spot Which countries are most frequently visited by tourists from other countries? The table shows results according to Travel and Leisure magazine ( 2005\()\). a. Is country visited a categorical or a quantitative variable? b. In creating a bar graph of these data, would it be most sensible to list the countries alphabetically or in the form of a Pareto chart? Explain. c. Does either a dot plot or stem-and-leaf plot make sense for these data? Explain. $$ \begin{array}{lc} \hline {\text { Most Visited Countries, } 2005} \\ \hline \text { Country } & \text { Number of Visits (millions) } \\ \hline \text { France } & 77.0 \\ \text { China } & 53.4 \\ \text { Spain } & 51.8 \\ \text { United States } & 41.9 \\ \text { Italy } & 39.8 \\ \text { United Kingdom } & 24.2 \\ \text { Canada } & 20.1 \\ \text { Mexico } & 19.7 \\ \hline \end{array} $$

European fertility The European fertility rates (mean number of children per adult woman) from Exercise 2.18 are shown again in the table. a. Find the median of the fertility rates. Interpret. b. Find the mean of the fertility rates. Interpret. c. For each woman, the number of children is a whole number, such as 2 or 3 . Explain why it makes sense to measure a mean number of children per adult woman (which is not a whole number) to compare fertility levels, such as the fertility levels of 1.5 in Canada and 2.4 in Mexico. $$\begin{array}{lclc}\hline \text { Country } & \text { Fertility } & \text { Country } & \text { Fertility } \\ \hline \text { Austria } & 1.4 & \text { Netherlands } & 1.7 \\ \text { Belgium } & 1.7 & \text { Norway } & 1.8 \\ \text { Denmark } & 1.8 & \text { Spain } & 1.3 \\ \text { Finland } & 1.7 & \text { Sweden } & 1.6 \\ \text { France } & 1.9 & \text { Switzerland } & 1.4 \\ \text { Germany } & 1.3 & \text { United Kingdom } & 1.7 \\ \text { Greece } & 1.3 & \text { United States } & 2.0 \\ \text { Ireland } & 1.9 & \text { Canada } & 1.5 \\ \text { Italy } & 1.3 & \text { Mexico } & 2.4 \\ \hline\end{array}$$

In New Zealand, the mean and median weekly earnings for males in 2009 was \(\$ 993\) and \(\$ 870\), respectively and for females, the mean and median weekly earnings were \(\$ 683\) and \(\$ 625\), respectively (www.nzdotstat.stats.govt.nz). Does this suggest that the distribution of weekly earnings for males is symmetric, skewed to the right, or skewed to the left? What about the distribution of weekly earnings for females? Explain.
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