Question
Graph 1
An unlabeled dot plot. The dot plot has arrows at both ends and ranges from 0 to 5 in increments of 1. There are 2 dots above 0, 4 dots above 1, 5 dots above 2, 3 dots above 3, 2 dots above 4, and 2 dots above 5.
Graph 2
An unlabeled dot plot.
Which of the two graphs has a larger standard deviation?
(1 point)
Responses
The standard deviation is the same for both graphs.
The standard deviation is the same for both graphs.
The first graph has a larger standard deviation.
The first graph has a larger standard deviation.
The second graph has a larger standard deviation.
The second graph has a larger standard deviation.
It is impossible to tell from this data.
An unlabeled dot plot. The dot plot has arrows at both ends and ranges from 0 to 5 in increments of 1. There are 2 dots above 0, 4 dots above 1, 5 dots above 2, 3 dots above 3, 2 dots above 4, and 2 dots above 5.
Graph 2
An unlabeled dot plot.
Which of the two graphs has a larger standard deviation?
(1 point)
Responses
The standard deviation is the same for both graphs.
The standard deviation is the same for both graphs.
The first graph has a larger standard deviation.
The first graph has a larger standard deviation.
The second graph has a larger standard deviation.
The second graph has a larger standard deviation.
It is impossible to tell from this data.
Answers
GPT-4o mini
To determine which of the two graphs has a larger standard deviation, we can analyze the data given for Graph 1 and consider what we might know or assume about Graph 2.
### Graph 1 Data
- 0: 2 dots
- 1: 4 dots
- 2: 5 dots
- 3: 3 dots
- 4: 2 dots
- 5: 2 dots
#### Calculation of Mean for Graph 1
1. **Calculate the total number of dots:**
- Total = 2 + 4 + 5 + 3 + 2 + 2 = 18
2. **Calculate the mean (average):**
- Mean = (0*2 + 1*4 + 2*5 + 3*3 + 4*2 + 5*2) / 18
- = (0 + 4 + 10 + 9 + 8 + 10) / 18 = 41 / 18 ≈ 2.28
#### Calculation of Variance and Standard Deviation for Graph 1
3. **Calculate variance:**
Each value's squared deviation from the mean (∑(x - mean)² * frequency):
- (0 - 2.28)² * 2 = 10.4384 * 2 ≈ 20.8768
- (1 - 2.28)² * 4 = 1.6384 * 4 = 6.5536
- (2 - 2.28)² * 5 = 0.0784 * 5 = 0.392
- (3 - 2.28)² * 3 = 0.5184 * 3 = 1.5552
- (4 - 2.28)² * 2 = 2.9536 * 2 = 5.9072
- (5 - 2.28)² * 2 = 7.3924 * 2 = 14.7848
Sum of squared deviations:
- Total = 20.8768 + 6.5536 + 0.392 + 1.5552 + 5.9072 + 14.7848 ≈ 49.0696
Variance = Total / (N - 1) = 49.0696 / 17 ≈ 2.89
Standard deviation = √Variance ≈ √2.89 ≈ 1.70
### Graph 2 Analysis
We have no specific data for Graph 2, making it impossible to calculate a mean, variance, or standard deviation without knowing the distribution of the data, which might include:
- More concentrated around the mean (lower standard deviation)
- More spread out (higher standard deviation)
### Conclusion
Since we cannot compare Graph 2 with the determined values from Graph 1, we must conclude:
**It is impossible to tell from this data.**
Therefore, the correct response is:
**It is impossible to tell from this data.**
### Graph 1 Data
- 0: 2 dots
- 1: 4 dots
- 2: 5 dots
- 3: 3 dots
- 4: 2 dots
- 5: 2 dots
#### Calculation of Mean for Graph 1
1. **Calculate the total number of dots:**
- Total = 2 + 4 + 5 + 3 + 2 + 2 = 18
2. **Calculate the mean (average):**
- Mean = (0*2 + 1*4 + 2*5 + 3*3 + 4*2 + 5*2) / 18
- = (0 + 4 + 10 + 9 + 8 + 10) / 18 = 41 / 18 ≈ 2.28
#### Calculation of Variance and Standard Deviation for Graph 1
3. **Calculate variance:**
Each value's squared deviation from the mean (∑(x - mean)² * frequency):
- (0 - 2.28)² * 2 = 10.4384 * 2 ≈ 20.8768
- (1 - 2.28)² * 4 = 1.6384 * 4 = 6.5536
- (2 - 2.28)² * 5 = 0.0784 * 5 = 0.392
- (3 - 2.28)² * 3 = 0.5184 * 3 = 1.5552
- (4 - 2.28)² * 2 = 2.9536 * 2 = 5.9072
- (5 - 2.28)² * 2 = 7.3924 * 2 = 14.7848
Sum of squared deviations:
- Total = 20.8768 + 6.5536 + 0.392 + 1.5552 + 5.9072 + 14.7848 ≈ 49.0696
Variance = Total / (N - 1) = 49.0696 / 17 ≈ 2.89
Standard deviation = √Variance ≈ √2.89 ≈ 1.70
### Graph 2 Analysis
We have no specific data for Graph 2, making it impossible to calculate a mean, variance, or standard deviation without knowing the distribution of the data, which might include:
- More concentrated around the mean (lower standard deviation)
- More spread out (higher standard deviation)
### Conclusion
Since we cannot compare Graph 2 with the determined values from Graph 1, we must conclude:
**It is impossible to tell from this data.**
Therefore, the correct response is:
**It is impossible to tell from this data.**