Question

Here are the summary statistics for the weekly payroll of a small company: lowest salary \(=\$ 300\), mean salary \(=\$ 700\), median \(=\$ 500\), range \(=\$ 1200, \mathrm{IQR}=\) \(\$ 600\), first quartile \(=\$ 350\), standard deviation \(=\$ 400\). a) Do you think the distribution of salaries is symmetric, skewed to the left, or skewed to the right? Explain why. b) Between what two values are the middle \(50 \%\) of the salaries found? c) Suppose business has been good and the company gives every employee a \(\$ 50\) raise. Tell the new value of each of the summary statistics. d) Instead, suppose the company gives each employee a \(10 \%\) raise. Tell the new value of each of the summary statistics.

Short answer

a) Skewed to the right; b) Between \(\$350\) and \(\$950\); c) All are increased by \(\$50\); d) All are multiplied by 1.10.

Step-by-step solution

Analyzing Symmetry

To determine if the distribution is symmetric or skewed, compare the mean and median values. A symmetric distribution generally has the mean equal to the median. Here, the mean salary is \(\\(700\) and the median is \(\\)500\). Since the mean is considerably higher than the median, the distribution is likely skewed to the right.

Calculating the Middle 50%

The interquartile range (IQR) represents the range of the middle 50% of the data. It is calculated as \(Q3 - Q1\). Given the IQR is \(\\(600\) and the first quartile (Q1) is \(\\)350\), we find the third quartile (Q3) as \(Q1 + \text{IQR} = \\(350 + \\)600 = \\(950\). Therefore, the middle 50% of salaries fall between \(\\)350\) and \(\$950\).

Adjusting for a $50 Raise

When every employee receives a \(\\(50\) raise, all positions and measures of central tendency increase by \(\\)50\). New minimum salary = \(\\(300 + \\)50 = \\(350\); mean = \(\\)700 + \\(50 = \\)750\); median = \(\\(500 + \\)50 = \\(550\); max = original max + \(\\)50\); range remains \(\\(1200\) because both min and max increase by \(\\)50\); IQR remains \(\\(600\); Q1 = \(\\)350 + \\(50 = \\)400\); standard deviation remains \(\$400\) because it's not affected by a constant addition.

Adjusting for a 10% Raise

A 10% increase multiplies each salary by 1.10. New minimum salary = \(\\(300 \times 1.10 = \\)330\); mean = \(\\(700 \times 1.10 = \\)770\); median = \(\\(500 \times 1.10 = \\)550\); maximum salary = original max \(\times 1.10\); range = original range \(\times 1.10 = \\(1200 \times 1.10 = \\)1320\); IQR = original IQR \(\times 1.10 = \\(600 \times 1.10 = \\)660\); Q1 = \(\\(350 \times 1.10 = \\)385\); standard deviation = \(\\(400 \times 1.10 = \\)440\).

Key concepts

Understanding Mean and Median

The mean and median are two important measures of central tendency. They help us understand the general size of data and how it is distributed.

**Mean**: The mean is what most people refer to as the "average." It's calculated by adding up all the numbers and then dividing by the count of the numbers. In the context of salaries, it gives us an idea of the "average" salary an employee would earn if the total payroll was evenly split among all employees. In the given problem, the mean salary is $700, which suggests that if all salaries were distributed equally, each person would earn $700.

**Median**: The median is the middle number in a sorted list of numbers. If the list has an odd number of observations, the median is the middle one. If it has an even number, it's the average of the two middle numbers. The median salary indicates the point at which half of the salaries are higher and half are lower. Here, the median is $500, meaning half the employees earn less than $500, and half earn more.

When the mean is higher than the median, it typically means that some higher salaries are pulling the mean up. This difference hints at a skewed distribution.

Grasping Interquartile Range

The interquartile range (IQR) is a measure that tells us about the spread or variability of the middle half of a dataset. It's a part of descriptive statistics that highlights the range where most of your data points lie.

**Understanding IQR**: To calculate the IQR, you first need to determine the first quartile (Q1) and the third quartile (Q3). The IQR is then the difference between Q3 and Q1: \[ \text{IQR} = Q3 - Q1 \]

In the provided scenario, we know the IQR is \(600 and Q1 is \)350. By simply doing the math, we find that Q3 is \[ 350 + 600 = 950 \].

**Importance of IQR**: The IQR is not affected by very high or very low values (outliers), which makes it a reliable measure of spread. In this exercise, the middle 50% of all salaries fall between \(350 and \)950. This tells us that most employees earn salaries within this range, which can be quite informative for understanding salary distribution.

Exploring Skewness

Skewness is a measure that describes the symmetry or asymmetry of a distribution. It's one of the ways we can determine how balanced a set of data is around its mean.

**Understanding Skewness**: A distribution can be described as symmetrical, right-skewed, or left-skewed.

• In a **symmetrical distribution**, the mean and median are approximately equal. The left and right tails are mirror images.
• In a **right-skewed (positively skewed) distribution**, the mean is usually greater than the median. This skewness suggests that there are more values concentrated on the lower end while the high values create a longer tail on the right side.
• In a **left-skewed (negatively skewed) distribution**, the mean is typically less than the median. Here, the tail extends more towards the lower end.

For the payroll of the small company, since the mean salary is $700 and the median is $500, the distribution is likely right-skewed. This indicates that a few employees earn significantly more than the rest, pulling the mean upward.

Delving into Standard Deviation

Standard deviation is a widely used measure of spread or dispersion in a dataset. It provides insight into how close or spread apart the data points are around the mean.

**Calculating Standard Deviation**: Calculating the standard deviation involves several steps, often requiring a bit of math. However, the important thing to remember is that it provides an average distance of each data point from the mean.

**Interpreting Standard Deviation**: A smaller standard deviation suggests that the data points are closer to the mean, indicating consistency. Conversely, a larger standard deviation indicates more spread out data, showing more variability. For the company, a standard deviation of $400 signifies a moderate level of salary dispersion.

**Impact of Changes**: When a constant is added to each data point, like a $50 raise for each employee, the standard deviation remains unchanged because the spread of data doesn't change. However, if every data point is multiplied by a constant, like a 10% raise, the standard deviation changes proportionally. In this example, a 10% raise increases the standard deviation from $400 to $440, demonstrating greater variability due to the increase in salaries.