Question

Given the following data set: .23, .30, .35, .41 , .56, .58, .76, .80 a. Find the lower and upper quartiles. b. Calculate the IQR. c. Calculate the lower and upper fences. Are there any outliers?

Short answer

Answer: In the given dataset, the lower quartile Q1 = .325, upper quartile Q3 = .67, the interquartile range (IQR) = .345, and there are no outliers.

Step-by-step solution

1. Arrange the data in ascending order

The given data set is: .23, .30, .35, .41, .56, .58, .76, .80 Next, we need to find the lower quartile (Q1) and upper quartile (Q3).

2. Find the lower quartile (Q1)

To find Q1, we take the middle value of the lower half of the data set. Since we have 8 data points, the lower half includes the first 4 numbers: .23, .30, .35, .41. The middle value between .30 and .35 is .325, so Q1 = .325

3. Find the upper quartile (Q3)

To find Q3, we take the middle value of the upper half of the data set. The upper half includes the last 4 numbers: .56, .58, .76, .80. The middle value between .58 and .76 is .67, so Q3 = .67 Now, we can find the interquartile range (IQR).

4. Calculate the interquartile range (IQR)

To find the IQR, subtract Q1 from Q3: IQR = Q3 - Q1 = .67 - .325 = .345 Then, we will calculate the lower and upper fences.

5. Calculate the lower and upper fences

To find the lower fence, subtract 1.5 times the IQR from Q1: Lower fence = Q1 -1.5*IQR = .325 - 1.5*.345 = .1795 To find the upper fence, add 1.5 times the IQR to Q3: Upper fence = Q3 + 1.5*IQR = .67 + 1.5*.345 = .8155 Finally, let's identify any outliers in the data set.

6. Identify any outliers

An outlier is any data point below the lower fence or above the upper fence. The lower fence is .1795, and the upper fence is .8155. None of the data points fall outside this range, so there are no outliers in this data set. In summary: a. The lower quartile Q1 = .325, and the upper quartile Q3 = .67 b. The interquartile range (IQR) = .345 c. The lower fence = .1795, and the upper fence = .8155. There are no outliers in this data set.

Key concepts

Quartiles

Quartiles are helpful in understanding the spread and structure of a data set. Essentially, they divide the data into four equal parts. Let's go over how to find them.

- **Lower Quartile (Q1):** This is the median of the first half of the data. It represents the 25th percentile of the data set. For our example, with data points .23, .30, .35, and .41 as the lower half, we calculate Q1 as the middle value: - Arrange the first four numbers in ascending order. - Since four numbers are even, average the two middle numbers, .30 and .35. - Thus, Q1 = (.30 + .35)/2 = 0.325.
- **Upper Quartile (Q3):** This is the median of the second half of the data, showcasing the 75th percentile. - The last four numbers are .56, .58, .76, and .80. - Average the two middle values, .58 and .76 to find Q3. - Therefore, Q3 = (.58 + .76)/2 = 0.67. Understanding Q1 and Q3 helps in analyzing how the data is spread around the median. It provides insights into the data distribution. This can assist in identifying any unusual behaviors or patterns.

Interquartile Range (IQR)

The Interquartile Range (IQR) is a measure of statistical dispersion. It calculates the range within the middle 50% of a data set, allowing us to understand variability. Here's how you get it:

1. **Formula:** - Subtract the lower quartile (Q1) from the upper quartile (Q3). - The formula is: IQR = Q3 - Q1
2. **Calculation:** - For our data, we found Q3 = 0.67 and Q1 = 0.325. - Therefore, IQR = 0.67 - 0.325 = 0.345.
Having this middle range helps quantify how spread out the central data points are. It dismisses outliers, focusing on the core set. If the IQR is large, the data points are more spread out, while a smaller IQR suggests they are closer together.

Outlier Detection

Outliers in data can skew results and analyses, rendering statistical calculations misleading. That's why detecting them is crucial. Outlier detection often revolves around finding data points that significantly deviate from the other points.

1. **Fences Calculation:** - Use the IQR to set boundaries, known as fences. - Lower fence = Q1 - 1.5 * IQR - Upper fence = Q3 + 1.5 * IQR
2. **Identify Outliers:** - For the given data set: - Lower fence = 0.325 - 1.5 * 0.345 = 0.1795 - Upper fence = 0.67 + 1.5 * 0.345 = 0.8155
3. **Analysis:** - Any data point below 0.1795 or above 0.8155 would be classified as an outlier. - In this particular example, all data points lie within this range, hence there are no outliers. Detecting outliers is beneficial for getting cleaner data and more accurate statistical results. It refines the focus on the "true" data core, dismissing anomalies or errors.