Question
The owner of a bagel shop recorded the number of bagels sold each day for a week. Her results were 699, 751, 810, 842, 923, 1,061, 1,214. Describe the spread of number of bagels sold each day using the mean absolute deviation.
Answers
GPT-4o mini
To calculate the Mean Absolute Deviation (MAD) of the number of bagels sold, we will follow these steps:
1. **Calculate the Mean**: First, we need to find the mean (average) number of bagels sold over the week.
\[
\text{Mean} = \frac{\text{Sum of all bagels sold}}{\text{Number of days}}
\]
First, sum the number of bagels sold:
\[
699 + 751 + 810 + 842 + 923 + 1061 + 1214 = 5260
\]
Now, divide by the number of days (7):
\[
\text{Mean} = \frac{5260}{7} \approx 751.43
\]
2. **Calculate the Absolute Deviations**: Next, we calculate the absolute deviation of each day's sales from the mean.
\[
\begin{align*}
|699 - 751.43| & = 52.43 \\
|751 - 751.43| & = 0.43 \\
|810 - 751.43| & = 58.57 \\
|842 - 751.43| & = 90.57 \\
|923 - 751.43| & = 171.57 \\
|1061 - 751.43| & = 309.57 \\
|1214 - 751.43| & = 462.57 \\
\end{align*}
\]
3. **Sum of Absolute Deviations**:
Now we add these absolute deviations together:
\[
52.43 + 0.43 + 58.57 + 90.57 + 171.57 + 309.57 + 462.57 = 1095.14
\]
4. **Calculate the Mean Absolute Deviation**:
Finally, we divide the sum of the absolute deviations by the number of days (7):
\[
\text{MAD} = \frac{1095.14}{7} \approx 156.44
\]
### Summary
The Mean Absolute Deviation (MAD) of the number of bagels sold each day is approximately **156.44**. This value indicates the average distance of daily sales from the mean, which gives a sense of how spread out the daily bagel sales are around the average.
1. **Calculate the Mean**: First, we need to find the mean (average) number of bagels sold over the week.
\[
\text{Mean} = \frac{\text{Sum of all bagels sold}}{\text{Number of days}}
\]
First, sum the number of bagels sold:
\[
699 + 751 + 810 + 842 + 923 + 1061 + 1214 = 5260
\]
Now, divide by the number of days (7):
\[
\text{Mean} = \frac{5260}{7} \approx 751.43
\]
2. **Calculate the Absolute Deviations**: Next, we calculate the absolute deviation of each day's sales from the mean.
\[
\begin{align*}
|699 - 751.43| & = 52.43 \\
|751 - 751.43| & = 0.43 \\
|810 - 751.43| & = 58.57 \\
|842 - 751.43| & = 90.57 \\
|923 - 751.43| & = 171.57 \\
|1061 - 751.43| & = 309.57 \\
|1214 - 751.43| & = 462.57 \\
\end{align*}
\]
3. **Sum of Absolute Deviations**:
Now we add these absolute deviations together:
\[
52.43 + 0.43 + 58.57 + 90.57 + 171.57 + 309.57 + 462.57 = 1095.14
\]
4. **Calculate the Mean Absolute Deviation**:
Finally, we divide the sum of the absolute deviations by the number of days (7):
\[
\text{MAD} = \frac{1095.14}{7} \approx 156.44
\]
### Summary
The Mean Absolute Deviation (MAD) of the number of bagels sold each day is approximately **156.44**. This value indicates the average distance of daily sales from the mean, which gives a sense of how spread out the daily bagel sales are around the average.