Answer :
Let's tackle the question step-by-step, considering the given results for Group A and Group B.
Group A results: [0, 1, 2, 3, 5, 5, 7, 8, 9, 10]
Group B results: [5, 5, 5, 5, 5, 5, 5, 5, 5, 5]
### Part a) Calculate the mean, median, and mode for each group.
#### Mean
The mean is the average of the numbers in the group.
- Mean of Group A:
[tex]\[ \text{Mean}_A = \frac{0 + 1 + 2 + 3 + 5 + 5 + 7 + 8 + 9 + 10}{10} = \frac{50}{10} = 5.0 \][/tex]
- Mean of Group B:
[tex]\[ \text{Mean}_B = \frac{5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5}{10} = \frac{50}{10} = 5.0 \][/tex]
#### Median
The median is the middle value when the numbers are listed in order. If there is an even number of observations, the median is the average of the two middle numbers.
- Median of Group A:
[tex]\[ \text{Median}_A = \frac{5 + 5}{2} = 5.0 \][/tex]
- Median of Group B:
[tex]\[ \text{Median}_B = 5 \quad (\text{since all values are 5}) \][/tex]
#### Mode
The mode is the number that appears most frequently.
- Mode of Group A:
[tex]\[ \text{Mode}_A = 5 \quad (\text{appears twice}) \][/tex]
- Mode of Group B:
[tex]\[ \text{Mode}_B = 5 \quad (\text{appears ten times}) \][/tex]
To summarize, the mean, median, and mode for each group are as follows:
- Group A: Mean = 5.0, Median = 5.0, Mode = 5
- Group B: Mean = 5.0, Median = 5.0, Mode = 5
### Part b) Explain why none of the averages is useful for deciding which group is from which class.
Despite using different methods of calculating the average, all the averages (mean, median, and mode) for Group A and Group B are identical, i.e., 5.0, 5.0, and 5 respectively. This exact match of averages in both groups prevents us from distinguishing between the two groups based solely on these statistics. Therefore, these measures of central tendency do not help us determine which group is from the class set by ability and which is from the mixed-ability class.
### Part c) Explain why calculating the range is helpful in this case.
The range is the difference between the highest and lowest scores in a group. It gives us an idea of the spread or dispersion of the data.
- Range of Group A:
[tex]\[ \text{Range}_A = 10 - 0 = 10 \][/tex]
- Range of Group B:
[tex]\[ \text{Range}_B = 5 - 5 = 0 \][/tex]
The range for Group A (10) is much larger than the range for Group B (0). This indicates that the scores in Group A are more spread out, whereas all the scores in Group B are the same. This difference in range can be useful for distinguishing which group could be from a class set by ability and which is from a mixed-ability class. Typically, a mixed-ability class is likely to produce results that are more varied (wider range), while a class set by ability may produce more uniform results (narrower range). Based on this, Group A is likely to be the mixed-ability class and Group B the set by ability.
Group A results: [0, 1, 2, 3, 5, 5, 7, 8, 9, 10]
Group B results: [5, 5, 5, 5, 5, 5, 5, 5, 5, 5]
### Part a) Calculate the mean, median, and mode for each group.
#### Mean
The mean is the average of the numbers in the group.
- Mean of Group A:
[tex]\[ \text{Mean}_A = \frac{0 + 1 + 2 + 3 + 5 + 5 + 7 + 8 + 9 + 10}{10} = \frac{50}{10} = 5.0 \][/tex]
- Mean of Group B:
[tex]\[ \text{Mean}_B = \frac{5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5}{10} = \frac{50}{10} = 5.0 \][/tex]
#### Median
The median is the middle value when the numbers are listed in order. If there is an even number of observations, the median is the average of the two middle numbers.
- Median of Group A:
[tex]\[ \text{Median}_A = \frac{5 + 5}{2} = 5.0 \][/tex]
- Median of Group B:
[tex]\[ \text{Median}_B = 5 \quad (\text{since all values are 5}) \][/tex]
#### Mode
The mode is the number that appears most frequently.
- Mode of Group A:
[tex]\[ \text{Mode}_A = 5 \quad (\text{appears twice}) \][/tex]
- Mode of Group B:
[tex]\[ \text{Mode}_B = 5 \quad (\text{appears ten times}) \][/tex]
To summarize, the mean, median, and mode for each group are as follows:
- Group A: Mean = 5.0, Median = 5.0, Mode = 5
- Group B: Mean = 5.0, Median = 5.0, Mode = 5
### Part b) Explain why none of the averages is useful for deciding which group is from which class.
Despite using different methods of calculating the average, all the averages (mean, median, and mode) for Group A and Group B are identical, i.e., 5.0, 5.0, and 5 respectively. This exact match of averages in both groups prevents us from distinguishing between the two groups based solely on these statistics. Therefore, these measures of central tendency do not help us determine which group is from the class set by ability and which is from the mixed-ability class.
### Part c) Explain why calculating the range is helpful in this case.
The range is the difference between the highest and lowest scores in a group. It gives us an idea of the spread or dispersion of the data.
- Range of Group A:
[tex]\[ \text{Range}_A = 10 - 0 = 10 \][/tex]
- Range of Group B:
[tex]\[ \text{Range}_B = 5 - 5 = 0 \][/tex]
The range for Group A (10) is much larger than the range for Group B (0). This indicates that the scores in Group A are more spread out, whereas all the scores in Group B are the same. This difference in range can be useful for distinguishing which group could be from a class set by ability and which is from a mixed-ability class. Typically, a mixed-ability class is likely to produce results that are more varied (wider range), while a class set by ability may produce more uniform results (narrower range). Based on this, Group A is likely to be the mixed-ability class and Group B the set by ability.
