Problem 57 - Basic Statistics for Business and Economics with Student

Question

The accounting firm of Crawford and Associates has five senior partners. Yesterday the senior partners saw six, four, three, seven, and five clients, respectively. a. Compute the mean number and median number of clients seen by a partner. b. Is the mean a sample mean or a population mean? Why? c. Verify that \(\Sigma(X-\mu)=0\).

Step-by-Step Solution

Answer

a. Mean = 5, Median = 5; b. Population mean; c. Verified, \(\Sigma(X-\mu)=0\).

Step 1: List the Number of Clients

List out the number of clients each partner saw: 6, 4, 3, 7, and 5.

Step 2: Calculate the Mean

The mean number of clients, \( \mu \), is calculated by summing the total number of clients and dividing by the number of partners. So, \( \mu = \frac{6 + 4 + 3 + 7 + 5}{5} = \frac{25}{5} = 5 \).

Step 3: Order the Data for Median

To find the median, order the data set from least to greatest: 3, 4, 5, 6, 7.

Step 4: Determine the Median

The median is the middle value of the ordered data set. Here, the median is 5.

Step 5: Identify Mean Type

Since we have all the data of the group of senior partners (which consists of 5 partners), the mean calculated is a population mean, not a sample mean.

Step 6: Calculate \(\Sigma(X-\mu)\)

Calculate \( \Sigma(X-\mu) \) by finding \((X - \mu)\) for each partner's client count: \( (6-5), (4-5), (3-5), (7-5), (5-5) \). This results in: 1, -1, -2, 2, 0. Sum these values to verify: \( 1 + (-1) + (-2) + 2 + 0 = 0 \).

Key Concepts

Mean Calculation

The mean, often referred to as the average, provides a central value for a data set, signifying the sum of all data points divided by the number of observations. In our scenario, the mean number of clients seen by each partner is calculated by adding the client counts: 6, 4, 3, 7, and 5. This gives us a total of 25 clients.
To find the mean, we divide the total number of clients by the number of partners, which is 5. Thus, the mean, represented as \( \mu \), is \( \mu = \frac{25}{5} = 5 \).

• Mean tells us the overall average performance of the partners
• It's useful for determining an average trend in a data set

Understanding how to calculate and interpret the mean is fundamental in descriptive statistics and helps in comparing different sets of data.

Median Calculation

The median is the middle value of an ordered data set. It provides another measure of central tendency, which is less sensitive to extreme values than the mean. To determine the median, first, we need to sort the partner's client counts in ascending order: 3, 4, 5, 6, 7.
The number of data points (clients seen) is 5, which is odd, so the median is the third value in this ordered list. Hence, the median number of clients seen is 5.

• The median is particularly helpful when dealing with skewed distributions or outliers
• Unlike the mean, it is not affected by extremely high or low values

Mastering the concept of median calculation provides a clearer insight into data distribution itself.

Population Mean vs. Sample Mean

Understanding the distinction between population mean and sample mean is crucial in statistics. A population mean is calculated when all data points of a particular group are present and accounted for. Conversely, a sample mean is derived from a subset of the broader population data.
In our case, since we have data from all five senior partners, the mean calculated is a population mean, denoted as \( \mu \).

• Population mean represents the "true" mean for the entire group
• Sample mean, while similar in calculation, often includes a margin for error or estimation error

This distinction is essential when interpreting data results and reporting findings based on the data being reviewed.

Summation Notation

Summation notation, often represented by the symbol \( \Sigma \), is a concise way of expressing the sum of a series of terms. It's widely used in statistics to summarize various data set calculations. In this particular exercise, we examine \( \Sigma(X - \mu) \), which represents the sum of deviations of each data point \( X \) from the mean \( \mu \).
Calculating \( \Sigma(X - \mu) \), we determine the deviation for each partner's client count from the mean. These are 1, -1, -2, 2, and 0, respectively. Summing these deviations results in zero: \( 1 + (-1) + (-2) + 2 + 0 = 0 \). This verifies a fundamental property of means—when deviations from the mean are summed, they equal zero.

• Summation notation simplifies complex calculations into understandable expressions
• It is fundamental for deriving other statistical measures, such as variance and standard deviation

Grasping this notation helps streamline your ability to work with large data sets effectively.