Question

Searching online for the price of an item you would like to buy can save you quite a bit of money. A search for the best price for a white KitchenAid 5-speed hand mixer using yroo.com lists 16 sellers with various prices found in the following table. $$\begin{array}{lclc}\hline \text { Seller } & \text { Price (\$) } & \text { Seller } & \text { Price (\$) } \\\\\hline \text { Best Buy } & 39.99 & \text { Sears } & 39.99 \\\\\text { Blain's Farm \& Fleet } & 39.99 & \text { Boston Store } & 39.99 \\\\\text { Bergners } & 39.99 & \text { Home Depot } & 41.46 \\\\\text { Wayfair } & 42.46 & \text { Buydig.com } & 43.99 \\\\\text { Target } & 46.49 & \text { Shopko } & 47.49 \\\\\text { Jet.com } & 47.90 & \text { Amazon } & 47.19 \\\\\text { Walmart } & 51.97 & \text { Office Depot } & 54.99 \\\\\text { Houzz } & 59.99 & \text { True Value } & 59.99 \\\\\hline\end{array}$$ a. What is the average price of this hand mixer for these 16 sellers? b. What is the median price for these 16 sellers? c. As a consumer would you be interested in the average price of the hand mixer? The median price? What other descriptive measures would be important to you?

Short answer

#tag_title#Step 2: Find the sum of the prices#tag_content#Add up all the prices: $$ \begin{aligned} &39.99 + 39.99 + 39.99 + 39.99 + 39.99 + 41.46 + 42.46 + 43.99 \\ + &46.49 + 47.49 + 47.90 + 47.19 + 51.97 + 54.99 + 59.99 + 59.99 \end{aligned} $$ The sum is $763.88$. #tag_title#Step 3: Calculate the mean#tag_content#Divide the sum by the number of prices (16), to get the mean: $$ \text{Mean} = \frac{763.88}{16} = 47.74 $$ The mean price is $47.74$. b. Calculate the median price: #tag_title#Step 1: Arrange the prices in increasing order#tag_content#List the prices in ascending order: $$ \begin{array}{l} 39.99, 39.99, 39.99, 39.99, 39.99, 41.46, 42.46, 43.99,\\ 46.49, 47.49, 47.90, 47.19, 51.97, 54.99, 59.99, 59.99 \end{array} $$ The prices are already listed in increasing order. #tag_title#Step 2: Determine the median#tag_content#Since there are 16 prices (an even number), the median is the average of the two middle values: the 8th and the 9th prices. In this case, those prices are $43.99$ and $46.49$. $$ \text{Median} = \frac{43.99 + 46.49}{2} = \frac{90.48}{2} = 45.24 $$ The median price is $45.24$. Short answer: The mean price for a hand mixer is $47.74$, and the median price is $45.24$.

Step-by-step solution

List the prices

List all the prices given in the table: $$ \begin{array}{l} 39.99, 39.99, 39.99, 39.99, 39.99, 41.46, 42.46, 43.99,\\ 46.49, 47.49, 47.90, 47.19, 51.97, 54.99, 59.99, 59.99 \end{array} $$

Key concepts

Average Price Calculation

When shopping online or analyzing market prices, determining the average price of a product is a crucial step. The average, also known as the mean, is computed by summing up all the individual prices and then dividing by the total number of prices.

To calculate the average price for the 16 sellers listed, one would add each price and then divide the total by 16. Mathematically, the average price (AP) can be expressed as:
\[ AP = \frac{{\sum_{i=1}^{n} P_i}}{n} \]
where \( P_i \) represents the price from each seller and \( n \) is the number of sellers. By performing this calculation, buyers get an idea of what they might expect to pay on average and can assess if a given price is above or below this benchmark.

Median Price Calculation

The median price is another critical statistic that offers a different insight into pricing data. It represents the middle value of a sorted list of numbers—meaning half the prices are above it and half below. To find the median, one must first arrange the prices in ascending order and then locate the central number.

If there is an odd number of prices, the median is simply the middle value. However, if there is an even number of prices, as is the case with the 16 sellers, the median is the average of the two central numbers. This statistical measure is particularly useful for understanding the distribution of prices and identifying outliers, as the median is not affected by extremely high or low values unlike the average.

Statistical Measures

In addition to average and median prices, other statistical measures provide valuable insights when analyzing data. These include:

• Mode: The most frequently occurring price in the dataset.
• Range: The difference between the highest and the lowest price, which indicates price variability.
• Quartiles: Values that divide the dataset into quarters, showing the spread of the middle 50% of prices.
• Variance and Standard Deviation: Indicators of how spread out the prices are around the average.

Each of these measures can help consumers understand different aspects of pricing, such as consistency among sellers or the likelihood of finding a significant bargain.

Data Analysis

Data analysis involves examining, cleaning, transforming, and modeling data to gain insights and inform decision-making. In the context of price comparison, shoppers may use data analysis techniques to identify trends, outliers, and statistical measures that can influence their purchasing decisions.

A thorough data analysis could reveal if a low price is a typical bargain or a too-good-to-be-true offer indicating a potential compromise on quality or service. Similarly, exploring how the prices relate to each other and to additional variables like shipping costs, seller rating, or warranty terms adds depth to understanding what constitutes a good deal.