Question

A clerk entering salary data into a company spreadsheet accidentally put an extra " \(0^{\prime \prime}\) in the boss's salary, listing it as \(\$ 2,000,000\) instead of \(\$ 200,000 .\) Explain how this error will affect these summary statistics for the company payroll: a) measures of center: median and mean. b) measures of spread: range, IQR, and standard deviation.

Short answer

The error increases the mean, range, and standard deviation but usually does not affect the median or IQR.

Step-by-step solution

Analyze the Impact on the Median

The median is the middle value in a data set when the values are arranged in order. The error in the boss's salary doesn’t change the position of the median unless this salary is near the middle of the salary list. Therefore, if other salaries remain the same and the number is not in the middle of the dataset, the median remains unchanged.

Analyze the Impact on the Mean

The mean is calculated by adding all the salaries together and dividing by the number of salaries. The error in the boss's salary significantly increases the total sum, thus affecting the mean by increasing it. As this value is a tenfold error, the mean moves notably upwards.

Analyze the Impact on the Range

The range is calculated as the difference between the highest and lowest salary. Since \(\\(2,000,000\) is significantly higher than the correct salary \(\\)200,000\), it will create a much larger range than accurate salary data would.

Analyze the Impact on the IQR

The Interquartile Range (IQR) is calculated as the difference between the 75th percentile (Q3) and the 25th percentile (Q1) of the dataset. If the erroneous salary value, \(\$2,000,000\), is an outlier and does not affect Q1 or Q3, then the IQR remains unchanged because it focuses on the middle 50% of the data.

Analyze the Impact on the Standard Deviation

The standard deviation measures how spread out the numbers are from the mean. The large error in the boss’s salary significantly increases the spread of the data, hence significantly increasing the standard deviation.

Key concepts

Measures of Center

Measures of center like the median and mean help us understand where the middle of a data set lies. They act as a quick summary of data and are crucial for comparative analysis.
The **median** is the middle number in a data set organized in ascending order. It divides the data into two equal parts. Depending on the number of data points, it may even result in the average of the two middle values.
Meanwhile, the **mean** provides the average value by summing up all the individual data points and dividing by the total number of items. The mean is sensitive to extreme values, commonly referred to as outliers, and can be greatly affected by them.

Measures of Spread

Measures of spread, like range, IQR, and standard deviation, describe the variability within a data set. These metrics show how much the values in a dataset differ from one another and the mean.
The **range** shows variability by subtracting the smallest data point from the largest. A larger range indicates more spread within the dataset.
The **Interquartile Range (IQR)** measures the range within the middle 50% of the data, calculated by subtracting the first quartile (Q1) from the third quartile (Q3). It provides insight into the consistency of the middle section of the dataset.
The **standard deviation** provides an average distance of each data point from the mean. A higher standard deviation indicates more variability within the dataset, while a lower value points to less spread.

Impact on Median and Mean

When an error, such as an inflated salary, occurs, it affects the median and mean differently.
The **median** remains relatively stable against such errors unless the mistake lands directly in the center of the arranged data points. This means that, in most cases, the median won't change significantly with an error in an extreme value like the boss's salary.
The **mean**, on the other hand, is highly sensitive and notably impacted by extreme values. In our example, misreporting the boss’s salary as $2,000,000 instead of $200,000 drastically increases the average salary calculation. This happens because the mean is calculated by adding all salaries together so that a larger number dramatically skews the total.

Impact on Range and IQR

Instances of error in data entry can distort measures of spread such as range and IQR significantly.
**Range** is directly affected since it depends on both the maximum and minimum values. Thus, an erroneous entry like a tenfold increase in salary increases the range disproportionately to reflect a larger spread.
However, the **Interquartile Range (IQR)** is usually unaffected by outliers like an inflated salary as this measure disregards extreme values. It’s focused strictly within the dataset's middle half. Only if the outliers alter Q1 or Q3, the IQR might change, but such influence is rare with individual anomalies.

Impact on Standard Deviation

The **standard deviation** is heavily influenced when errors introduce extremities into data. Such outliers like an incorrect boss's salary at $2,000,000 significantly elevate how scattered the dataset is around its mean.
This occurs as standard deviation reflects the dispersion of dataset entries from the calculated mean. When the mean itself is skewed upwards by extreme values, the deviation from which every other value is measured increases. Ultimately, large errors cause the standard deviation to report a much higher variability level in the data than what might be accurate.