Question

Use these \(n=24\) measurements to answer the questions in Exercises \(9-10 .\) \(\begin{array}{llllll}4.5 & 3.2 & 3.5 & 3.9 & 3.5 & 3.9 \\ 4.3 & 4.8 & 3.6 & 3.3 & 4.3 & 4.2 \\ 3.9 & 3.7 & 4.3 & 4.4 & 3.4 & 4.2 \\ 4.4 & 4.0 & 3.6 & 3.5 & 3.9 & 4.0\end{array}\) Find the sample mean and median.

Short answer

Answer: The sample mean is approximately 3.7792, and the median is 3.95.

Step-by-step solution

Understand the Data

Analyze the given set of 24 measurements. The data is presented in the following manner: \(\begin{array}{llllll}4.5 & 3.2 & 3.5 & 3.9 & 3.5 & 3.9 \\\ 4.3 & 4.8 & 3.6 & 3.3 & 4.3 & 4.2 \\\ 3.9 & 3.7 & 4.3 & 4.4 & 3.4 & 4.2 \\\ 4.4 & 4.0 & 3.6 & 3.5 & 3.9 & 4.0\end{array}\)

Calculate the Sample Mean

To calculate the sample mean, sum up all the measurements and divide by the total number of measurements (\(n = 24\)). \(\bar{x} = \frac{\sum x_{i}}{n}\) \(\bar{x} = \frac{4.5+3.2+3.5+3.9+3.5+3.9+4.3+4.8+3.6+3.3+4.3+4.2+3.9+3.7+4.3+4.4+3.4+4.2+4.4+4.0+3.6+3.5+3.9+4.0}{24}\) \(\bar{x} = \frac{90.7}{24}\) \(\bar{x} = 3.7792\) So, the sample mean is approximately \(3.7792\).

Calculate the Median

In order to calculate the median, first reorder the measurements from least to greatest. Then find the middle value(s). Reordered measurements: \(3.2, 3.3, 3.4, 3.5, 3.5, 3.6, 3.6, 3.7, 3.9, 3.9, 3.9, 3.9, 4.0, 4.0, 4.2, 4.2, 4.3, 4.3, 4.3, 4.4, 4.4, 4.5, 4.8\) Since there are 24 measurements (an even number), the median will be the average of the two middle values (12th and 13th values). Median \(= \frac{3.9 + 4.0}{2}\) Median \(= 3.95\) The median of the given measurements is \(3.95\).

Key concepts

Sample Mean Calculation

Understanding the sample mean is crucial when delving into the world of descriptive statistics. It represents the average value in a set of numbers, which gives us a quick summary of the dataset. Beginning with a collection of values, such as the 24 measurements provided, the first step is to sum all the individual values, which is symbolically represented as \( \sum x_i \). After obtaining the sum, you need to divide it by the number of measurements present in your sample, in this case, 24.

To put it plainly, if you had scored 24 different marks in several tests, and you wanted to estimate your overall performance, you would calculate the average mark. This average, or sample mean, gives a concise representation of your scores in a single number. It's a method of condensing the information of all your test scores into a digestible snapshot. In statistical terms, the calculation results in what is known as the sample mean, symbolized as \( \bar{x} \) and is given by the formula:
\[\bar{x} = \frac{\sum x_{i}}{n}\]
In this specific exercise, adding up the given measurements and using the formula, the sample mean is approximately 3.7792. This indicates that on average, the measurements center around this value.

Median Calculation

The median, another fundamental concept in descriptive statistics, is the middle value of a data set that has been arranged in ascending order. It quite literally stands at the center of your data, dividing it into two halves. To find the median of a dataset, you first organize the numbers from the smallest to the largest. After you've done that, if your dataset has an odd number of values, the median is the number right in the middle. If there's an even number of values, like in our example with 24 measurements, you take the average of the two central numbers.

Imagine having all your friends lined up by height. The median would be the height of the friend standing in the middle of this line-up. If there is an even number of friends, you would find the average height between the two centermost friends. Similarly, the median calculation for our given dataset is done by taking the average of the 12th and 13th measurements when aligned from least to greatest, resulting in a median of 3.95. This tells us that half of the measurements are below or equal to 3.95 and the other half are above or equal to 3.95.

Data Analysis

Data analysis is the process of systematically examining data to glean meaningful information. In the realm of statistics, this often involves using measures like the sample mean and median. However, data analysis doesn't stop there: one must also consider the spread of the data, such as how broadly the measurements vary, and look for any patterns or outliers that could impact the results.

Interpreting the Results

When we analyze data, we're not just crunching numbers; we're interpreting what those numbers tell us about the world. For instance, if we're looking at temperatures within a month, the mean gives us an indication of the overall warmth while the median can provide insight into the typical daily temperature.

Beyond Central Tendency

In our exercise, the sample mean and median provide a snapshot of central tendency, but real insight comes from diving deeper. Measures like variance and standard deviation will let us understand the variability or consistency among the measurements. This comprehensive analysis is vital for making informed decisions based on the data, whether it's assessing the overall performance of a manufacturing process or understanding trends in environmental data.