1 answer
(click or scroll down)
Bot GPT 4-o mini To determine the "usual" values in the data set and identify any outliers, we can use the Interquartile Range (IQR) method. The steps to find the limits are as follows:
1. **Order the Data**: Sort the data from smallest to largest.
```
20, 20, 21, 22, 23, 24, 25, 25, 26, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29, 29, 30, 31, 31, 31, 31, 32, 33, 33, 33, 33, 34, 34, 34, 34, 35, 35, 35, 35, 36, 36, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 40, 41, 42, 62
```
2. **Calculate Q1 (First Quartile)**: This is the median of the first half of the data (the lower half).
- The lower half: \(20, 20, 21, 22, 23, 24, 25, 25, 26, 27, 27, 27, 28, 28\)
- Median of the lower half (Q1) = Average of 25th and 26th values = (25 + 26) / 2 = 25.5.
3. **Calculate Q3 (Third Quartile)**: This is the median of the second half of the data (the upper half).
- The upper half: \(28, 28, 29, 29, 29, 29, 30, 31, 31, 31, 31, 32, 33, 33, 33, 34, 34, 34, 34, 35, 35, 35, 35, 36, 36, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 40, 41, 42, 62\)
- Median of the upper half (Q3) = Average of 35th and 36th values = (35 + 36) / 2 = 35.5.
4. **Calculate IQR**:
\[
IQR = Q3 - Q1 = 35.5 - 25.5 = 10.
\]
5. **Determine Outlier Limits**:
- Lower limit: \( Q1 - 1.5 \times IQR = 25.5 - 1.5 \times 10 = 25.5 - 15 = 10.5. \)
- Upper limit: \( Q3 + 1.5 \times IQR = 35.5 + 1.5 \times 10 = 35.5 + 15 = 50.5. \)
Thus, the usual range of values is between **10.5** and **50.5**.
6. **Identify Outliers**: Any value outside of these limits is considered an outlier.
- Outliers are: \( 62 \) (as it is greater than 50.5).
In conclusion:
- The limits for usual values are **10.5** to **50.5**.
- There is **1 outlier** in the data set (the value **62**).