Question
Here are the statistics for the high temperatures in a city during October:
mean of 65.3 degrees Fahrenheit
median of 63.5 degrees Fahrenheit
standard deviation of 9.3 degrees Fahrenheit
IQR of 7.1 degrees Fahrenheit
Recall that the temperature , measured in degrees Celsius, is related to the temperature , measured in degrees Fahrenheit, by .
Describe how the value of each statistic changes when 32 is subtracted from the temperature in degrees Fahrenheit.
Describe how the value of each statistic further changes when the new values are multiplied by .
Describe how to find the value of each statistic when the temperature is measured in degrees Celsius.
mean of 65.3 degrees Fahrenheit
median of 63.5 degrees Fahrenheit
standard deviation of 9.3 degrees Fahrenheit
IQR of 7.1 degrees Fahrenheit
Recall that the temperature , measured in degrees Celsius, is related to the temperature , measured in degrees Fahrenheit, by .
Describe how the value of each statistic changes when 32 is subtracted from the temperature in degrees Fahrenheit.
Describe how the value of each statistic further changes when the new values are multiplied by .
Describe how to find the value of each statistic when the temperature is measured in degrees Celsius.
Answers
GPT-4o mini
To analyze the changes in the statistics with respect to the conversion from Fahrenheit to Celsius, let's take the following steps:
1. **Subtraction of 32 (Conversion from Fahrenheit to Fahrenheit - 32)**:
When you subtract 32 from the temperature measured in degrees Fahrenheit:
- **Mean**: The mean will decrease by 32. So, new mean = 65.3 - 32 = 33.3.
- **Median**: The median will also decrease by 32. New median = 63.5 - 32 = 31.5.
- **Standard Deviation**: The standard deviation will not change, because subtracting a constant (32) does not affect the spread of the data. It remains 9.3.
- **IQR**: The IQR will not change for the same reason as the standard deviation; it remains 7.1.
2. **Multiplication by \(\frac{5}{9}\) (Conversion from Fahrenheit to Celsius)**:
After subtracting 32, we will multiply the adjusted Fahrenheit values by \(\frac{5}{9}\) to convert them to Celsius:
- **Mean**: New mean = (33.3) * \(\frac{5}{9}\) ≈ 18.5.
- **Median**: New median = (31.5) * \(\frac{5}{9}\) ≈ 17.5.
- **Standard Deviation**: The standard deviation will also be multiplied by \(\frac{5}{9}\). New standard deviation = (9.3) * \(\frac{5}{9}\) ≈ 5.2 (rounded to one decimal place).
- **IQR**: The IQR will be multiplied by \(\frac{5}{9}\) as well. New IQR = (7.1) * \(\frac{5}{9}\) ≈ 3.9 (rounded to one decimal place).
3. **Summary of the statistics in Celsius**:
After these transformations (subtracting 32, followed by multiplying by \(\frac{5}{9}\)), the new statistics in degrees Celsius are:
- Mean: Approximately 18.5 degrees Celsius
- Median: Approximately 17.5 degrees Celsius
- Standard Deviation: Approximately 5.2 degrees Celsius
- IQR: Approximately 3.9 degrees Celsius
To summarize:
- Subtracting 32 shifts all values down by 32 degrees.
- Multiplying by \(\frac{5}{9}\) scales the values to Celsius and affects the mean, median, standard deviation, and IQR accordingly.
1. **Subtraction of 32 (Conversion from Fahrenheit to Fahrenheit - 32)**:
When you subtract 32 from the temperature measured in degrees Fahrenheit:
- **Mean**: The mean will decrease by 32. So, new mean = 65.3 - 32 = 33.3.
- **Median**: The median will also decrease by 32. New median = 63.5 - 32 = 31.5.
- **Standard Deviation**: The standard deviation will not change, because subtracting a constant (32) does not affect the spread of the data. It remains 9.3.
- **IQR**: The IQR will not change for the same reason as the standard deviation; it remains 7.1.
2. **Multiplication by \(\frac{5}{9}\) (Conversion from Fahrenheit to Celsius)**:
After subtracting 32, we will multiply the adjusted Fahrenheit values by \(\frac{5}{9}\) to convert them to Celsius:
- **Mean**: New mean = (33.3) * \(\frac{5}{9}\) ≈ 18.5.
- **Median**: New median = (31.5) * \(\frac{5}{9}\) ≈ 17.5.
- **Standard Deviation**: The standard deviation will also be multiplied by \(\frac{5}{9}\). New standard deviation = (9.3) * \(\frac{5}{9}\) ≈ 5.2 (rounded to one decimal place).
- **IQR**: The IQR will be multiplied by \(\frac{5}{9}\) as well. New IQR = (7.1) * \(\frac{5}{9}\) ≈ 3.9 (rounded to one decimal place).
3. **Summary of the statistics in Celsius**:
After these transformations (subtracting 32, followed by multiplying by \(\frac{5}{9}\)), the new statistics in degrees Celsius are:
- Mean: Approximately 18.5 degrees Celsius
- Median: Approximately 17.5 degrees Celsius
- Standard Deviation: Approximately 5.2 degrees Celsius
- IQR: Approximately 3.9 degrees Celsius
To summarize:
- Subtracting 32 shifts all values down by 32 degrees.
- Multiplying by \(\frac{5}{9}\) scales the values to Celsius and affects the mean, median, standard deviation, and IQR accordingly.