Question

For the data sets in Exercises \(13-15,\) find the mean, the median, and the mode. Comment on the skewness or symmetry of the data. The following data give the estimated prices of a 170-gram can or a 200-gram pouch of water-packed tuna for 14 different brands, based on prices paid nationally in supermarkets. \(\begin{array}{rrrrrrr}.99 & 1.92 & 1.23 & .85 & .65 & .53 & 1.41 \\ 1.12 & .63 & .67 & .69 & .60 & .60 & .66\end{array}\)

Short answer

Answer: The mean of the tuna brand prices is approximately 0.9286, the median is 0.68, and the mode is 0.60. The data is right-skewed (positively skewed) since the mean is greater than the median and the mode.

Step-by-step solution

Order the data in ascending order

Arrange the prices from lowest to highest. The data will look like this: \(.53, .60, .60, .63, .65, .66, .67, .69, .85, .99, 1.12, 1.23, 1.41, 1.92\)

Calculate the mean

To find the mean, sum up all the prices and divide by the number of data points. With 14 data points, the mean can be calculated as follows: \(\text{Mean} = \frac{.53 + .60 + .60 + .63 + .65 + .66 + .67 + .69 + .85 + .99 + 1.12 + 1.23 + 1.41 + 1.92}{14} \approx 0.9286\)

Calculate the median

To find the median, locate the middle value in the ordered data set. Since we have an even number of data points, we need to take the average of the two middle values, which are the 7th and 8th values in this case: \(\text{Median} = \frac{.67 + .69}{2} = 0.68\)

Calculate the mode

The mode is the value that occurs most often in the data set. In our case, the value \(.60\) occurs twice. So the mode is: \(\text{Mode} = .60\)

Comment on the skewness or symmetry

We have the mean (\(\approx 0.9286\)), the median (\(0.68\)) and the mode (\(.60\)). Since the mean is greater than the median and the mode, the data is right-skewed (positively skewed). If the mean, median, and mode were close or equal to each other, the data would be symmetrical.

Key concepts

Mean Calculation

The mean, often referred to as the average, is a critical measure in descriptive statistics. It is calculated by adding up all the values in a data set and then dividing by the number of values. In the context of our exercise, where we analyze the prices of tuna across different brands, the mean gives us an overall estimation of the typical cost consumers might expect.

To calculate the mean price, we add every individual price and then divide by the total number of brands, which is 14 in this case. The mean provides a single summary statistic that represents the center of the data. However, it's important to note that the mean can be affected by outliers, or values that are much lower or higher than the rest of the data set.

Median Calculation

The median is the 'middle' value of a data set when it's arranged in ascending or descending order. If the data set has an odd number of values, the median is the value exactly in the middle. For even-numbered data sets, like in our exercise with tuna prices, the median is the average of the two middle values.

After ordering the data from the lowest to the highest price, we determine the two central prices and calculate their average. This process gives us the median, which is very useful, especially when we deal with skewed data because it is not as influenced by outliers as the mean is. The median divides the data set into two equal parts and represents a value that separates the higher half from the lower half of the data points.

Mode Calculation

The mode in a set of data is the value that appears most frequently. It's quite possible for a data set to have more than one mode if multiple values occur with the same frequency or to have no mode at all if all values are unique. In our exercise, the mode is determined by identifying the price that shows up most often among the different tuna brands.

This measure is particularly significant when data is qualitative. When dealing with prices like in our case, the mode can inform us about the most common cost customers might pay for a product. It's also beneficial for understanding the data's distribution and any potential for clustering around specific values.

Data Skewness Analysis

Skewness describes the asymmetry of the distribution of values in a data set. There are two types of skewness: right (positive) skewness where the tail of the distribution is longer on the right side, and left (negative) skewness where it extends more to the left.

In our analysis, we observed that the mean is higher than both the median and the mode, indicating a right skew. This means that a few higher-priced tuna brands pull the mean upward, whereas most brands are clustered around the lower prices. Skewness analysis is invaluable for understanding the spread and outliers within a data set and can influence decision-making processes by highlighting where most of the data points lie.

Probability and Statistics

The realm of probability and statistics provides tools to analyze and make inferences about data. While probability deals with predicting the likelihood of future events, statistics involves the analysis of the frequency of past events. Mean, median, and mode are foundational concepts within statistics that allow us to summarize and interpret data more efficiently.

Grasping these measures helps us to describe the characteristics of a data set, as we demonstrated with our tuna price example. In the broader scope of probability and statistics, these metrics are used to make predictions, inform decisions, and understand patterns within various fields — from business and economics to healthcare and education. They form the basis upon which more complex statistical analyses are built, playing an essential role in the interpretation of quantitative research.