A wide variety of oak trees grow in the United States. In one study, a sample of acorns was collected from different locations, and their volumes, in cm3, were recorded. In the table are summary statistics for these data:
N 38
Mean 3.0
Median 1.8
St. Dev. 2.6
Minimum 0.3
Maximum 10.5
1st Quartile 1.1
3rd Quartile 4.3
Describe a procedure that uses some or all of these summary statistics to determine whether outliers are present in the data.
Since the mean is larger than the median, the distribution is positively skewed. The mean ± 2 standard deviations covers about 95% of scores in a normal distribution. That should start you going.
To determine whether outliers are present in the data, you can follow the steps below using the summary statistics provided:
Step 1: Understand the concept of outliers
Outliers are data points that are significantly different from the other data points in a dataset. They can distort the overall statistical analysis and should be investigated further to ensure data integrity.
Step 2: Calculate the Interquartile Range (IQR)
IQR is the range between the 1st quartile (Q1) and the 3rd quartile (Q3). It gives a measure of the spread of the middle 50% of the data.
IQR = Q3 - Q1
Using the given data,
IQR = 4.3 - 1.1 = 3.2
Step 3: Determine the lower and upper boundaries for identifying outliers
The lower boundary (LB) is calculated as:
LB = Q1 - 1.5 * IQR
The upper boundary (UB) is calculated as:
UB = Q3 + 1.5 * IQR
Using the values from the given data:
LB = 1.1 - 1.5 * 3.2 = -3.3
UB = 4.3 + 1.5 * 3.2 = 9.7
Step 4: Identify outliers
Any data point that falls below the lower boundary (LB) or above the upper boundary (UB) is considered an outlier.
Based on the given data, the minimum value is 0.3, which falls above the lower boundary (-3.3), so it is not considered an outlier. However, the maximum value of 10.5 falls above the upper boundary (9.7), indicating that it may be an outlier.
Step 5: Further investigation
If there are potential outliers, it is recommended to examine them individually, considering their context and potential reasons for being different from the rest of the data. It may be necessary to check for data collection errors, verify measurement methods, or consult domain experts to understand the outliers better.
Remember that the presence of outliers may impact the analysis and conclusions drawn from the data, so they should be handled appropriately based on the specific context of the study.
To determine whether outliers are present in the data, you can use the summary statistics provided. Here's a step-by-step procedure using the summary statistics:
1. Identify the Interquartile Range (IQR): The IQR is a measure of the spread of the middle 50% of the data. It can be calculated by subtracting the 1st quartile (Q1) from the 3rd quartile (Q3). In this case, IQR = Q3 - Q1.
2. Calculate the Lower and Upper Bound: The lower bound is calculated as Q1 minus 1.5 times the IQR, and the upper bound is calculated as Q3 plus 1.5 times the IQR. These bounds help identify values that are considered outliers.
Lower Bound = Q1 - 1.5 * IQR
Upper Bound = Q3 + 1.5 * IQR
3. Identify any values outside the Lower and Upper Bound: Compare each value in the dataset with the lower and upper bound. If a value is below the lower bound or above the upper bound, it can be considered an outlier.
4. Visualize the data: Create a box plot to visually represent the data and inspect for any points that fall outside the whiskers (which represent the lower and upper bounds).
Using this procedure, outliers can be identified by comparing each value in the dataset with the lower and upper bounds, calculated based on the quartiles.