Question

For the data in Exercises \(4-6,\) calculate the median and the upper and lower quartiles. \(n=7\) measurements: 6,7,3,2,8,10,4

Short answer

Answer: The median is 6, the lower quartile (Q1) is 3, and the upper quartile (Q3) is 8.

Step-by-step solution

Arrange the data in ascending order

We start by arranging the given data in ascending order: 2, 3, 4, 6, 7, 8, 10.

Calculate the median

In this step, we will find the middle value of the ordered data set. Since there are 7 measurements, the median is the 4th value. Median = 6.

Find the lower quartile (Q1)

To find the lower quartile, we need to find the median of the lower half of the data set. The lower half of the data set consists of the first 3 values: 2, 3, and 4. Q1 is the median of this lower half, which is the 2nd value in the ordered set. Q1 = 3.

Find the upper quartile (Q3)

To find the upper quartile, we need to find the median of the upper half of the data set. The upper half of the data set consists of the last 3 values: 7, 8, and 10. Q3 is the median of this upper half, which is the 2nd value in the ordered upper half data set. Q3 = 8. The median is 6, the lower quartile (Q1) is 3, and the upper quartile (Q3) is 8.

Key concepts

Calculating Median

The median is a crucial measure in descriptive statistics, serving as the middle value of a data set when it's arranged in ascending order. For odd-numbered sets, the median is simply the value that has an equal number of items above and below it. In our textbook problem, we had seven data points, so the median was the fourth value which is 6.

For an even number of data points, calculating the median requires a slightly different approach. In this case, you take the average of the two middle numbers. Understanding how to calculate the median correctly is essential, as it is a measure of central tendency that can give you a quick snapshot of the 'middle' of your data, often more representative of a 'typical' value than the mean when the distribution is skewed.

Data Arrangement

Data arrangement, also known as data sorting, is the process of organizing data in a specific order, usually ascending (from smallest to largest) or descending (from largest to smallest). This step is fundamental before calculating measures like medians and quartiles.

In our example, we arranged the data in ascending order which gave us 2, 3, 4, 6, 7, 8, 10. Sorting data simplifies finding the median, quartiles, and analyzing the distribution. A well-organized data set provides a clearer insight into the range and enables easier identification of statistical outliers, which might indicate an error in data collection or reveal a significant finding.

Statistics Exercises

Statistics exercises like the one we are discussing help students build essential skills in data analysis and interpretation. These exercises often focus on identifying key components of data sets, such as measures of central tendency (mean, median, mode) and measures of dispersion (range, variance, standard deviation).

Hands-on practice with data sets—arranging the data, identifying outliers, and calculating statistical measures—reinforces the concept learning. Importantly, these exercises are not just about getting the right answer, but also about understanding the process and being able to explain and justify the methods used.

Descriptive Statistics

Descriptive statistics summarize or describe the characteristics of a data set. This branch of statistics deals with measures of central tendency and measures of variability (also called dispersion). Measures of central tendency include the mean, median, and mode, while measures of variability include the range, interquartile range, variance, and standard deviation.

Our problem involves calculating the median and quartiles—key components of descriptive statistics. Understanding these concepts is vital because they provide a simple summary of the data. This summary can then be used for further analysis in inferential statistics, which will help with predictions and decision making based on data patterns.