Question
Number of goals scored in
one season at Whitmore
College soccer games
follows:
0, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3,
3, 4, 4, 4, 4
Which measure of
variability—IQR or MAD—
best describes the spread
of the dataset?
IQR; the data
distribution is
symmetrical
MAD; the data
distribution has
outliers
MAD; the data
distribution is
symmetrical
IQR; the data
distribution is
skewed
one season at Whitmore
College soccer games
follows:
0, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3,
3, 4, 4, 4, 4
Which measure of
variability—IQR or MAD—
best describes the spread
of the dataset?
IQR; the data
distribution is
symmetrical
MAD; the data
distribution has
outliers
MAD; the data
distribution is
symmetrical
IQR; the data
distribution is
skewed
Answers
GPT-4o mini
To determine which measure of variability—Interquartile Range (IQR) or Mean Absolute Deviation (MAD)—best describes the spread of the dataset for goals scored in Whitmore College soccer games, we first need to analyze the data:
The dataset is:
0, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4
### Step 1: Calculate the IQR
1. **Order the dataset** (already ordered):
- 0, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4
2. **Find Q1 (the first quartile)**:
- Q1 is the median of the first half of the data:
- First half = 0, 0, 2, 2, 2, 2, 2 (7 numbers)
- The median of this group (Q1) = 2
3. **Find Q3 (the third quartile)**:
- Q3 is the median of the second half of the data:
- Second half = 3, 3, 3, 3, 3, 4, 4, 4, 4 (7 numbers)
- The median of this group (Q3) = 3
4. **Calculate IQR**:
- IQR = Q3 - Q1 = 3 - 2 = 1
### Step 2: Calculate the MAD
1. **Find the mean of the data**:
- Mean = (0 + 0 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 4 + 4) / 16
- Mean = 2.5
2. **Calculate the absolute deviations from the mean**:
- |0 - 2.5| = 2.5
- |0 - 2.5| = 2.5
- |2 - 2.5| = 0.5
- |2 - 2.5| = 0.5
- |2 - 2.5| = 0.5
- |2 - 2.5| = 0.5
- |2 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |4 - 2.5| = 1.5
- |4 - 2.5| = 1.5
- |4 - 2.5| = 1.5
- |4 - 2.5| = 1.5
3. **Mean of these absolute deviations (MAD)**:
- Sum of absolute deviations = 2.5 + 2.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 1.5 + 1.5 + 1.5 + 1.5 = 12
- MAD = 12 / 16 = 0.75
### Step 3: Analyze the Dataset's Spread
Now that we have calculated both measures of variability:
- IQR = 1
- MAD = 0.75
The data distribution has repeated values mainly skewed towards the low side (0 and 2), and some higher values (3 and 4), but since there are no extreme outliers causing significant skew, we can conclude that the IQR is likely best suited for describing the spread in this context.
### Conclusion
The correct choice is:
- **IQR; the data distribution is skewed**
The dataset is:
0, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4
### Step 1: Calculate the IQR
1. **Order the dataset** (already ordered):
- 0, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4
2. **Find Q1 (the first quartile)**:
- Q1 is the median of the first half of the data:
- First half = 0, 0, 2, 2, 2, 2, 2 (7 numbers)
- The median of this group (Q1) = 2
3. **Find Q3 (the third quartile)**:
- Q3 is the median of the second half of the data:
- Second half = 3, 3, 3, 3, 3, 4, 4, 4, 4 (7 numbers)
- The median of this group (Q3) = 3
4. **Calculate IQR**:
- IQR = Q3 - Q1 = 3 - 2 = 1
### Step 2: Calculate the MAD
1. **Find the mean of the data**:
- Mean = (0 + 0 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + 4 + 4) / 16
- Mean = 2.5
2. **Calculate the absolute deviations from the mean**:
- |0 - 2.5| = 2.5
- |0 - 2.5| = 2.5
- |2 - 2.5| = 0.5
- |2 - 2.5| = 0.5
- |2 - 2.5| = 0.5
- |2 - 2.5| = 0.5
- |2 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |3 - 2.5| = 0.5
- |4 - 2.5| = 1.5
- |4 - 2.5| = 1.5
- |4 - 2.5| = 1.5
- |4 - 2.5| = 1.5
3. **Mean of these absolute deviations (MAD)**:
- Sum of absolute deviations = 2.5 + 2.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 1.5 + 1.5 + 1.5 + 1.5 = 12
- MAD = 12 / 16 = 0.75
### Step 3: Analyze the Dataset's Spread
Now that we have calculated both measures of variability:
- IQR = 1
- MAD = 0.75
The data distribution has repeated values mainly skewed towards the low side (0 and 2), and some higher values (3 and 4), but since there are no extreme outliers causing significant skew, we can conclude that the IQR is likely best suited for describing the spread in this context.
### Conclusion
The correct choice is:
- **IQR; the data distribution is skewed**