Question

A high school senior uses the Internet to get information on February temperatures in the town where hell be going to college. He finds a Web site with some statistics, but they are given in degrees Celsius. The conversion formula is \({ }^{\circ} \mathrm{F}=9 / 5^{\circ} \mathrm{C}+32\). Determine the Fahrenheit equivalents for the summary information below. Maximum temperature \(=11^{\circ} \mathrm{C} \quad\) Range \(=33^{\circ}\) Mean \(=1^{\circ} \quad\) Standard deviation \(=7^{\circ}\) Median \(=2^{\circ} \quad\) IQR \(=16^{\circ}\)

Short answer

Maximum: \(50.8^{\circ}F\); Range: \(59.4^{\circ}F\); Mean: \(33.8^{\circ}F\); Standard Deviation: \(12.6^{\circ}F\); Median: \(35.6^{\circ}F\); IQR: \(28.8^{\circ}F\).

Step-by-step solution

Understand the Conversion Formula

The conversion formula to change degrees Celsius to degrees Fahrenheit is given by \(^{\circ}F=\frac{9}{5}^{\circ}C+32\). We will use this formula to convert each of the statistical measures from Celsius to Fahrenheit.

Convert Maximum Temperature

Convert the maximum temperature of \(11^{\circ}C\) to Fahrenheit using the formula:\[^{\circ}F = \frac{9}{5} \times 11 + 32 = 19.8 + 32 = 50.8^{\circ}F\]

Convert Mean Temperature

Convert the mean temperature of \(1^{\circ}C\) to Fahrenheit:\[^{\circ}F = \frac{9}{5} \times 1 + 32 = 1.8 + 32 = 33.8^{\circ}F\]

Convert Median Temperature

Convert the median temperature of \(2^{\circ}C\) to Fahrenheit:\[^{\circ}F = \frac{9}{5} \times 2 + 32 = 3.6 + 32 = 35.6^{\circ}F\]

Convert Temperature Ranges and Deviations

For the range and standard deviation, which represent differences in temperature, we only multiply by \(\frac{9}{5}\), not add 32, because they aren't absolute temperatures:- Range \((33^{\circ}C)\): \[9/5 \times 33 = 59.4^{\circ}F\]- Standard deviation \((7^{\circ}C)\): \[9/5 \times 7 = 12.6^{\circ}F\]

Convert Interquartile Range (IQR)

Similarly, convert the interquartile range \((16^{\circ}C)\):\[9/5 \times 16 = 28.8^{\circ}F\]

Key concepts

Celsius to Fahrenheit

Temperature conversion is essential, especially when dealing with international data presented in different temperature scales. The primary formula to convert a temperature from degrees Celsius to degrees Fahrenheit is:

• \(^{\circ}F = \frac{9}{5} \times ^{\circ}C + 32\)

Understanding this conversion is crucial for accurately interpreting data. The formula consists of two main components: the multiplication factor \(\frac{9}{5}\) and the addition of 32.
The multiplication factor accounts for the scale difference between Celsius and Fahrenheit. Celsius uses 100 degrees between the freezing point and boiling point of water, compared to Fahrenheit which uses 180 degrees in the same range. The addition of 32 bridges the starting points of the two scales: 0°C is equivalent to 32°F.
It's important to note that when converting differences, such as ranges or deviations, you only multiply by \(\frac{9}{5}\) without adding 32. This is because they measure relative change, not absolute temperature points.

Statistical Measures

Statistical measures like mean, median, range, standard deviation, and interquartile range (IQR) help us understand data sets in a more structured way.
Here's a brief explanation of each:

• **Mean**: The average of all data points. Here it gives you a general idea of the central tendency of temperatures.
• **Median**: The middle value when all data points are arranged in order. It is useful for understanding the central tendency when your data includes outliers.
• **Range**: The difference between the highest and lowest values. It indicates the spread of temperatures.
• **Standard Deviation**: Reflects how much individual data points deviate from the mean. A large standard deviation means the data points are spread out over a large range of values.
• **Interquartile Range (IQR)**: The range between the first quartile (25th percentile) and the third quartile (75th percentile), showing the middle 50% of data.

These statistical measures allow us to understand data variability and central tendency, important for making sense of weather patterns.

Data Interpretation

Interpreting data goes beyond just understanding the numbers; it involves analyzing what they mean in context. When looking at temperature data, consider the implications for daily life or specific applications.
For instance, knowing the average maximum temperature helps people prepare for expected weather conditions, while understanding the range or standard deviation reflects potential variations. It supports planning outfits, equipment, or activities that are temperature-sensitive.
Additionally, interquartile ranges and medians provide insights into typical weather patterns, especially useful for those moving to new climates or planning travel. By converting and interpreting these statistics in the Fahrenheit scale, people accustomed to this scale can more easily relate the information to their own experiences.
Understanding both statistical measures and conversion processes is key for accurate data interpretation across different contexts and units.