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Chapter 3: Problem 50

In 2010 , the laborers of a major factory in Australia went on strike. At that time, the average salary of a laborer was AU\$ \(21.60\) per hour, and the median salary was AU\$ \(20.00\). If you were representing the owners, which summary would you use to convince the public that a strike was not needed? If you were representing the laborers, which would you use? Why was there a discrepancy between the mean and median hourly rates? Explain. (Source: www.payscale.com)

Short Answer

Expert verified
The factory owners would use the average wage (AU$21.60) to argue against the need for a strike, while the laborers would point to the median wage (AU$20.00) to justify their cause. The discrepancy between these two figures implies that wage increases may not be evenly distributed among the laborers, suggesting wage inequality.

Step by step solution

01

Understand the Mean and Median

The mean, or average, is found by adding up all the values and dividing by the total number of values. The median is the middle value in a set of values when arranged in ascending or descending order. If the mean is greater than the median as in this case, it suggests that the data is positively skewed, i.e., a few laborers are earning significantly more, thus increasing the average wage.
02

Representing the Owners

If representing the factory owners, it would be better to refer to the average salary, which is higher (AU$21.60). This might convince the public that the laborers are being paid a reasonable wage, thus a strike is unnecessary.
03

Representing the Laborers

If representing the laborers, it would be better to use the median salary, which is lower (AU$20.00). This indicates that half of all laborers earn AU$20.00 or less per hour, making the need for a strike more justifiable.
04

Understanding the Discrepancy

The discrepancy between the average and the median wage is likely due to wage inequality. If a small number of laborers are earning much more than the others, those 'outlier' wages can heavily influence the average, making it larger than the median. This suggests that wage increases may not be evenly distributed among the laborers.

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Key Concepts

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

Understanding Statistical Measures
When assessing numerical data, it's crucial to be familiar with statistical measures such as the mean (average) and the median (middle value). Mean is calculated by summing all the values in a set and dividing by the count of the values. In contrast, the median is the value that falls right in the middle of an ordered list from smallest to largest, or vice versa.

These measures are foundational in interpreting and comparing datasets. For example, in the context of salaries, the mean might be used to assess the overall level of pay, whereas the median provides insight into what a typical worker might earn, dismissing the effect of extreme values.
Analyzing Data Distribution
The data distribution shapes our understanding of a dataset's characteristics. In an evenly spread distribution, the mean and median will be very close or identical. However, the presence of outliers or a long tail in one direction causes the data to become skewed.

Positive skew, or right-skew, occurs when higher values stretch towards the right, lifting the mean above the median. Conversely, negative skew, or left-skew, is when lower values drag the mean below the median. Recognizing the type of skewness helps in choosing the appropriate measure (mean or median) to represent the typical value in the dataset.
The Issue of Wage Inequality
Wage inequality is a critical social and economic issue that can be illuminated through statistics. It refers to the disparity in earnings among individuals. When a small number of laborers earn much more than the rest, as in the exercise, it creates a gap鈥攖he average wage is pulled up by these higher earners, but the median wage, representative of the 'typical' worker, remains lower.

This dissonance can misrepresent the financial well-being of the workforce when only mean wages are reported. Hence, analyzing wage inequality requires looking at both mean and median wages to accurately understand the distribution of pay across all workers.
Interpreting Skewed Data
In the realm of statistics, skewed data can often provide a deceptive view of the underlying scenario if not accurately interpreted. Skewness affects the mean more than the median because it's sensitive to extreme values. In the labor scenario mentioned, a positively skewed wage distribution hints that a handful of laborers are earning substantially more than their peers.

Understanding that the data is skewed can prevent misleading conclusions and ensure that both perspectives鈥攎anagement's and laborers'鈥攁re appropriately represented. Skewness in data highlights the importance of considering multiple statistical measures to capture the full picture.

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Most popular questions from this chapter

The teachers of a school collected data on the self-reported numbers of sick leaves taken by students in the months of September and October. $$ \begin{aligned} &\text { September: } 8,3,2,1,6,4,1,8,2,2,2,5,7,1,8,9 \text { , } \\ &\qquad 4,3,2,6,8,1,3,9,8,1,1,12,9,10 \end{aligned} $$ $$ \begin{aligned} &\text { October: } 2,8,3,6,1,5,4,9,8,1,1,12,9,9,0,6,1,8 \text { , } \\ &4,2,1,9,12,8,1,3,2,4,7,1 \end{aligned} $$ a. Compare the means in a sentence or two. b. Compare the standard deviation in a sentence or two.

A father has five children of ages 2,3 , 5,8, and 9 years. a. Calculate the standard deviation of their current ages. b. Without doing any calculation, indicate whether the standard deviation of the children's ages in the next 15 years will be larger, smaller, or the same as the standard deviation of their current ages. Check your answers by calculating the standard deviation of the ages in 15 years. Explain how adding 15 to each number affects the standard deviation. c. Find the mean of the children at their current ages. d. Without doing any calculation, indicate whether the mean of the children's age in the next 15 years will be larger, smaller, or the same as the mean of the current ages. Confirm your answer, and describe how adding 15 to each number affects the mean.

Assume that men's shoe sizes have a mean of 7 and a standard deviation of \(1.5\). a. What men's shoe size corresponds to a \(z\) -score of \(1.00\) ? b. What men's shoe size corresponds to a \(z\) -score of \(-1.50\) ?

Distributions of delays in flight arrival of a particular airline are roughly bell-shaped. The mean delay in arrival time is 12 minutes, and the standard deviation is 4 minutes. Which is more usual, a flight arriving 4 minutes earlier than the mean delay time or a flight arriving 4 minutes after this time? Explain.

College students Diane Glover and Esmeralda Olguin asked 25 men and 25 women how many speeding tickets they had received in the last three years. Men: 14 men said they had 0 tickets, 9 said they had 1 ticket, 1 had 2 tickets, and 1 had 5 tickets. Women: 18 said they had 0 tickets, 6 said they had 1 ticket, and 1 said she had 2 tickets. Is there evidence that the men and women differed? Answer by making appropriate plots and comparing appropriate summary statistics. Be sure to comment on the shape of the distributions and to mention any unusual features you observe.
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