Question
Number of hours in one
week students on a girls’
high school field hockey
team spent exercising:
14, 14, 14, 14, 15, 15, 15,
16, 16, 16, 16, 16, 17, 17,
17, 18, 18, 18, 18, 18
Which measure of
variability—IQR or MAD—
best describes the spread
of the dataset?
IQR; the data
distribution is
skewed
MAD; the data
distribution is
symmetrical
IQR; the data
distribution is
symmetrical
MAD; the data
distribution is
skewed
week students on a girls’
high school field hockey
team spent exercising:
14, 14, 14, 14, 15, 15, 15,
16, 16, 16, 16, 16, 17, 17,
17, 18, 18, 18, 18, 18
Which measure of
variability—IQR or MAD—
best describes the spread
of the dataset?
IQR; the data
distribution is
skewed
MAD; the data
distribution is
symmetrical
IQR; the data
distribution is
symmetrical
MAD; the data
distribution is
skewed
Answers
GPT-4o mini
To determine which measure of variability (IQR or MAD) best describes the spread of the given dataset, we first need to analyze the data.
### Step 1: Analyzing the Data
The dataset is:
14, 14, 14, 14, 15, 15, 15,
16, 16, 16, 16, 16, 17, 17,
17, 18, 18, 18, 18, 18
### Step 2: Calculating the IQR and MAD
1. **IQR (Interquartile Range)**:
- To find quartiles, we first need to arrange the data in order (it is already ordered).
- The median (Q2) of the dataset is the average of the 10th and 11th values:
Median = (16 + 16) / 2 = 16
- Lower quartile (Q1) is the median of the first half (first 10 values):
Q1 = (14 + 15) / 2 = 14.5
- Upper quartile (Q3) is the median of the second half (last 10 values):
Q3 = (17 + 18) / 2 = 17.5
- IQR = Q3 - Q1 = 17.5 - 14.5 = 3
2. **MAD (Mean Absolute Deviation)**:
- First, find the mean of the dataset:
Mean = (14*4 + 15*3 + 16*5 + 17*3 + 18*5) / 20 = 16.05
- Calculate the absolute deviations from the mean:
|14 - 16.05|, |14 - 16.05|, |14 - 16.05|, |14 - 16.05|, |15 - 16.05|, |15 - 16.05|, |15 - 16.05|,
|16 - 16.05|, |16 - 16.05|, |16 - 16.05|, |16 - 16.05|, |16 - 16.05|, |17 - 16.05|,
|17 - 16.05|, |17 - 16.05|, |18 - 16.05|, |18 - 16.05|, |18 - 16.05|, |18 - 16.05|, |18 - 16.05|
- Calculating the absolute deviations gives (the list of deviations is omitted for brevity, but we calculate them):
The mean of these absolute deviations gives us the MAD.
### Step 3: Determine the Nature of Distribution
- The data appears to cluster around the middle values (15, 16, and 17).
- To find whether it is skewed or symmetrical, we visualize (or think of) the distribution:
- It doesn't seem to lean heavily in one direction based on the values provided.
- However, there are more values on the lower end (14-15) compared to the higher end (17-18), hinting at a possible slight skew.
### Conclusion
- If the distribution was indeed symmetric, it would favor using MAD. However, given that there is a cluster of lower values and a tapering off on the high end, it might slightly favor skewing.
- Typically, IQR is a robust measure that handles skewed distributions better because it focuses on the middle 50% of the data.
Thus, the best option for your question seems to be:
**IQR; the data distribution is skewed**.
### Step 1: Analyzing the Data
The dataset is:
14, 14, 14, 14, 15, 15, 15,
16, 16, 16, 16, 16, 17, 17,
17, 18, 18, 18, 18, 18
### Step 2: Calculating the IQR and MAD
1. **IQR (Interquartile Range)**:
- To find quartiles, we first need to arrange the data in order (it is already ordered).
- The median (Q2) of the dataset is the average of the 10th and 11th values:
Median = (16 + 16) / 2 = 16
- Lower quartile (Q1) is the median of the first half (first 10 values):
Q1 = (14 + 15) / 2 = 14.5
- Upper quartile (Q3) is the median of the second half (last 10 values):
Q3 = (17 + 18) / 2 = 17.5
- IQR = Q3 - Q1 = 17.5 - 14.5 = 3
2. **MAD (Mean Absolute Deviation)**:
- First, find the mean of the dataset:
Mean = (14*4 + 15*3 + 16*5 + 17*3 + 18*5) / 20 = 16.05
- Calculate the absolute deviations from the mean:
|14 - 16.05|, |14 - 16.05|, |14 - 16.05|, |14 - 16.05|, |15 - 16.05|, |15 - 16.05|, |15 - 16.05|,
|16 - 16.05|, |16 - 16.05|, |16 - 16.05|, |16 - 16.05|, |16 - 16.05|, |17 - 16.05|,
|17 - 16.05|, |17 - 16.05|, |18 - 16.05|, |18 - 16.05|, |18 - 16.05|, |18 - 16.05|, |18 - 16.05|
- Calculating the absolute deviations gives (the list of deviations is omitted for brevity, but we calculate them):
The mean of these absolute deviations gives us the MAD.
### Step 3: Determine the Nature of Distribution
- The data appears to cluster around the middle values (15, 16, and 17).
- To find whether it is skewed or symmetrical, we visualize (or think of) the distribution:
- It doesn't seem to lean heavily in one direction based on the values provided.
- However, there are more values on the lower end (14-15) compared to the higher end (17-18), hinting at a possible slight skew.
### Conclusion
- If the distribution was indeed symmetric, it would favor using MAD. However, given that there is a cluster of lower values and a tapering off on the high end, it might slightly favor skewing.
- Typically, IQR is a robust measure that handles skewed distributions better because it focuses on the middle 50% of the data.
Thus, the best option for your question seems to be:
**IQR; the data distribution is skewed**.