Problem 58 - Basic Statistics for Business and Economics with Student

Question

Owens Orchards sells apples in a large bag by weight. A sample of seven bags contained the following numbers of apples: 23,19,26,17,21,24,22 a. Compute the mean number and median number of apples in a bag. b. Verify that \(\Sigma(x-\bar{x})=0\)

Step-by-Step Solution

Answer

Mean is 21.71, median is 22; sum of deviations is 0.

Step 1: Calculate the Mean

To find the mean, we sum all the numbers of apples in the bags and then divide by the number of observations. Add the numbers: \(23+19+26+17+21+24+22 = 152\). Then divide by the number of bags, which is 7: \(\frac{152}{7} = 21.71\). The mean number of apples is approximately 21.71.

Step 2: Arrange Data for Median Calculation

First, sort the sample data in ascending order: 17, 19, 21, 22, 23, 24, 26. The sorted data will help us to easily identify the median.

Step 3: Find the Median

Since there are 7 data points, the median will be the 4th number in this sorted list. Counting to the 4th number, we find that it is 22. Therefore, the median number of apples is 22.

Step 4: Calculate Deviations from the Mean

For each data point, subtract the mean (21.71) from the value: - \(23 - 21.71 = 1.29\)- \(19 - 21.71 = -2.71\)- \(26 - 21.71 = 4.29\)- \(17 - 21.71 = -4.71\)- \(21 - 21.71 = -0.71\)- \(24 - 21.71 = 2.29\)- \(22 - 21.71 = 0.29\)

Step 5: Verify Sum of Deviations

Add all the deviations: \(1.29 - 2.71 + 4.29 - 4.71 - 0.71 + 2.29 + 0.29 = 0\). The sum of the deviations is indeed 0, verifying that \(\Sigma(x-\bar{x}) = 0\).

Key Concepts

Mean Calculation

The mean, often called the average, is a key concept in descriptive statistics. It provides a central value of a data set. To find the mean, sum up all the values and divide by the number of observations. In our example with apple bags, the numbers are: 23, 19, 26, 17, 21, 24, and 22.
Here’s how we calculate the mean:

• Add all apple counts: 17 + 19 + 21 + 22 + 23 + 24 + 26 = 152.
• Divide by the total number of bags (7):\(\frac{152}{7} \approx 21.71\).

The mean number of apples per bag is approximately 21.71. This calculation simplifies understanding by providing a single measurement that reflects the overall data set.

Median Calculation

The median is a statistic that represents the middle value in a data set when the numbers are arranged in order. It's particularly useful in indicating the central tendency when data might be skewed by outliers. Here, we demonstrate finding the median using our apple count data.
Firstly, we must arrange the data in ascending order:

• Sorted apple counts: 17, 19, 21, 22, 23, 24, 26.

Since the sample size is odd (7 numbers), the median is the middle number. We count through the sorted list to the 4th number, which is 22.
So, the median number of apples per bag is 22. This value gives an idea of the central point in our dataset.

Sum of Deviations

The sum of deviations from the mean is a vital step in data analysis. It helps to check calculations and understand data spread. By definition, this is the sum of each data point's deviation from the mean value.Let's break down the deviations:

• Deviation for 23: 23 - 21.71 = 1.29
• Deviation for 19: 19 - 21.71 = -2.71
• Deviation for 26: 26 - 21.71 = 4.29
• Deviation for 17: 17 - 21.71 = -4.71
• Deviation for 21: 21 - 21.71 = -0.71
• Deviation for 24: 24 - 21.71 = 2.29
• Deviation for 22: 22 - 21.71 = 0.29

Adding these deviations gives:\[1.29 - 2.71 + 4.29 - 4.71 - 0.71 + 2.29 + 0.29 = 0\]This sum equals zero, which confirms the principle that the sum of deviations from the mean is always zero. This principle holds for any data set and is a fundamental aspect of mean calculations.