Question

Find the mean of the quartiles \(\mathrm{Q}_{1}, \mathrm{Q}_{2}\) and \(\mathrm{Q}_{3}\) of the data \(5,9,8,12,7,13,10,14\) (1) 9 (2) 10 (3) \(9.5\) (4) \(11.5\)

Short answer

A) 8.5 B) 9 C) 9.5 D) 10

Step-by-step solution

Arrange the data in ascending order

To sort the dataset in ascending order, reorder the numbers from smallest to largest: \(5,7,8,9,10,12,13,14\).

Identify Q1, Q2, and Q3 values

First, we need to identify the position of the quartiles: Q1 position: \(\frac{1}{4}(8)\) = 2 Q2 position: \(\frac{2}{4}(8)\) = 4 Q3 position: \(\frac{3}{4}(8)\) = 6 Now we can find the quartiles from the sorted dataset: Q1: the second value in the dataset, which is 7. Q2: the fourth value in the dataset, which is 9. Q3: the sixth value in the dataset, which is 12.

Calculate the mean of quartiles

To calculate the mean, sum the quartile values and divide the result by 3. So, using the formula mentioned above: Mean of quartiles = \(\frac{\mathrm{Q}_{1} + \mathrm{Q}_{2} + \mathrm{Q}_{3}}{3}\) = \(\frac{7+9+12}{3}\) = \(\frac{28}{3}\) = \(9\frac{1}{3}\) ≈ \(9.33\). Since answer options are given, we will round the mean to the nearest available option. Option (3) \(9.5\) is the closest answer.

Key concepts

Mean Calculation

Understanding mean calculation is essential for any statistics student. The mean, often referred to as the average, is a measure of central tendency, indicating the central or typical value in a set of data. To calculate the mean, you sum up all the values in your dataset and then divide by the number of values you have.

For instance, if we have a dataset with three values: 2, 4, and 6, the mean would be calculated as follows:

• Add the values together: 2 + 4 + 6 = 12.
• Divide by the number of values: 12 ÷ 3 = 4.

The mean, in this case, is 4. It is crucial for learners to keep in mind that the mean provides a quick snapshot of the data but can be influenced by very high or very low values, known as outliers. When dealing with quartiles, the 'mean of quartiles' is simply the arithmetic average of the three quartile values — a concept which helps us understand the spread of the central 50% of our data set.

Statistical Quartiles

Statistical quartiles are values that divide your data into quarters, once it's sorted in ascending order. They are used to give a quick and intuitive summary of the distribution of your data set. There are three main quartiles:

• First quartile (\(Q_1\)) is the median of the lower half of the dataset and marks the 25th percentile.
• Second quartile (\(Q_2\) or the median) divides the data set in half and marks the 50th percentile.
• Third quartile (\(Q_3\)) is the median of the upper half of the data set and marks the 75th percentile.

To find these values in a data set, one must first arrange the data in ascending order. Then, using a simple positional formula based on the number of data points, you can determine the position of each quartile and, subsequently, their values. In essence, these quartiles highlight the spread of the data, showing where the majority of data points lie and how they are skewed towards the lower or upper end of the scale.

Data Arrangement

Data arrangement is the first step taken when you're preparing to calculate statistical measures such as mean and quartiles. Organizing data into a specific order, typically in ascending or descending value, is crucial for many statistical calculations. In the context of quartiles, data must be listed in ascending order to accurately find the cut-off points that separate the data into quarters.

Once the data is arranged, it becomes easier to identify specific data positions and apply statistical rules. For example, to identify quartiles, you need to locate positions at the 25th, 50th, and 75th percentiles, which simply cannot be done without an ordered set. Proper data arrangement not only helps in finding quartiles, but it is also fundamental in identifying outliers, understanding the distribution of data, and in visualizing data through graphs like box plots which are heavily reliant on quartile information.