Answer :
Certainly! To understand which of these measures best summarizes Georgia's scores, let’s go through each option one by one and compute them step by step.
### 1. Mean
The mean is the average of all the scores. To calculate the mean:
[tex]\[ \text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n} \][/tex]
Where [tex]\( x_i \)[/tex] represents each score and [tex]\( n \)[/tex] represents the number of scores.
[tex]\[ \text{Mean} = \frac{89 + 75 + 92 + 68 + 78 + 75 + 80}{7} = \frac{557}{7} \approx 79.57 \][/tex]
### 2. Range
The range is the difference between the highest and lowest scores. To calculate the range:
[tex]\[ \text{Range} = \text{Max} - \text{Min} \][/tex]
[tex]\[ \text{Range} = 92 - 68 = 24 \][/tex]
### 3. Interquartile Range (IQR)
The IQR is the difference between the first quartile (Q1) and the third quartile (Q3). First, we need to sort the scores:
Sorted scores: [tex]\( 68, 75, 75, 78, 80, 89, 92 \)[/tex]
Next, we find the positions of Q1 and Q3. For our data set of 7 values:
- Q1 (25th percentile) is the value at position [tex]\( \frac{7 + 1}{4} = 2 \)[/tex]
- Q3 (75th percentile) is the value at position [tex]\( 3 \cdot \frac{7 + 1}{4} = 6 \)[/tex]
Checking the values:
- Q1 = 75
- Q3 = 89
Therefore, the IQR:
[tex]\[ \text{IQR} = \text{Q3} - \text{Q1} = 89 - 75 = 14 \][/tex]
### 4. Median
The median is the middle value when the data is sorted. If the number of observations is odd, it’s the middle number.
[tex]\[ \text{Median} = 78 \][/tex]
### 5. Mean Absolute Deviation (MAD)
The MAD is the average of the absolute differences between each score and the mean. First, let’s find these differences:
[tex]\[ |89 - 79.57| = 9.43 \][/tex]
[tex]\[ |75 - 79.57| = 4.57 \][/tex]
[tex]\[ |92 - 79.57| = 12.43 \][/tex]
[tex]\[ |68 - 79.57| = 11.57 \][/tex]
[tex]\[ |78 - 79.57| = 1.57 \][/tex]
[tex]\[ |75 - 79.57| = 4.57 \][/tex]
[tex]\[ |80 - 79.57| = 0.43 \][/tex]
Now calculate the mean of these absolute differences:
[tex]\[ \text{MAD} = \frac{9.43 + 4.57 + 12.43 + 11.57 + 1.57 + 4.57 + 0.43}{7} \approx 6.37 \][/tex]
### 6. Mode
The mode is the value that appears most frequently. In our data, 75 appears twice, and all other values appear once. So, the mode is:
[tex]\[ \text{Mode} = 75 \][/tex]
### Conclusion
Each of these measures provides a different type of summary:
- The mean gives an average score.
- The range measures the spread.
- The interquartile range measures the middle 50% spread.
- The median gives the middle score.
- The mean absolute deviation measures average deviation from the mean.
- The mode gives the most frequent score.
If Georgia wants a single value that summarizes all her scores, the mean (paying attention to all values) or the median (undisturbed by outliers) would typically serve as the best summaries. It depends on the specific context: whether she wants an average (mean) or a middle value (median). In practical application, Mean is more commonly used to summarize the dataset into a single value:
[tex]\[ \text{Mean} = 79.57 \][/tex]
### 1. Mean
The mean is the average of all the scores. To calculate the mean:
[tex]\[ \text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n} \][/tex]
Where [tex]\( x_i \)[/tex] represents each score and [tex]\( n \)[/tex] represents the number of scores.
[tex]\[ \text{Mean} = \frac{89 + 75 + 92 + 68 + 78 + 75 + 80}{7} = \frac{557}{7} \approx 79.57 \][/tex]
### 2. Range
The range is the difference between the highest and lowest scores. To calculate the range:
[tex]\[ \text{Range} = \text{Max} - \text{Min} \][/tex]
[tex]\[ \text{Range} = 92 - 68 = 24 \][/tex]
### 3. Interquartile Range (IQR)
The IQR is the difference between the first quartile (Q1) and the third quartile (Q3). First, we need to sort the scores:
Sorted scores: [tex]\( 68, 75, 75, 78, 80, 89, 92 \)[/tex]
Next, we find the positions of Q1 and Q3. For our data set of 7 values:
- Q1 (25th percentile) is the value at position [tex]\( \frac{7 + 1}{4} = 2 \)[/tex]
- Q3 (75th percentile) is the value at position [tex]\( 3 \cdot \frac{7 + 1}{4} = 6 \)[/tex]
Checking the values:
- Q1 = 75
- Q3 = 89
Therefore, the IQR:
[tex]\[ \text{IQR} = \text{Q3} - \text{Q1} = 89 - 75 = 14 \][/tex]
### 4. Median
The median is the middle value when the data is sorted. If the number of observations is odd, it’s the middle number.
[tex]\[ \text{Median} = 78 \][/tex]
### 5. Mean Absolute Deviation (MAD)
The MAD is the average of the absolute differences between each score and the mean. First, let’s find these differences:
[tex]\[ |89 - 79.57| = 9.43 \][/tex]
[tex]\[ |75 - 79.57| = 4.57 \][/tex]
[tex]\[ |92 - 79.57| = 12.43 \][/tex]
[tex]\[ |68 - 79.57| = 11.57 \][/tex]
[tex]\[ |78 - 79.57| = 1.57 \][/tex]
[tex]\[ |75 - 79.57| = 4.57 \][/tex]
[tex]\[ |80 - 79.57| = 0.43 \][/tex]
Now calculate the mean of these absolute differences:
[tex]\[ \text{MAD} = \frac{9.43 + 4.57 + 12.43 + 11.57 + 1.57 + 4.57 + 0.43}{7} \approx 6.37 \][/tex]
### 6. Mode
The mode is the value that appears most frequently. In our data, 75 appears twice, and all other values appear once. So, the mode is:
[tex]\[ \text{Mode} = 75 \][/tex]
### Conclusion
Each of these measures provides a different type of summary:
- The mean gives an average score.
- The range measures the spread.
- The interquartile range measures the middle 50% spread.
- The median gives the middle score.
- The mean absolute deviation measures average deviation from the mean.
- The mode gives the most frequent score.
If Georgia wants a single value that summarizes all her scores, the mean (paying attention to all values) or the median (undisturbed by outliers) would typically serve as the best summaries. It depends on the specific context: whether she wants an average (mean) or a middle value (median). In practical application, Mean is more commonly used to summarize the dataset into a single value:
[tex]\[ \text{Mean} = 79.57 \][/tex]
