91Ó°ÊÓ

The range is a basic way to understand the spread of data values in a set. It's simply the difference between the highest and lowest values. In our temperature data, we observe that the highest temperature is 23°F and the lowest is -7°F. To find the range, we perform the following calculation:
- Range = 23 - (-7) = 30
This means there is a spread of 30 degrees between the hottest and coldest days in our data set. Though easy to calculate, the range provides only a limited view of data distribution, as it doesn't account for how values are spread between the extremes.

Variance measures how much the numbers in a data set differ from the mean. It reflects the degree of spread in our temperature data. To calculate variance, we first subtract the mean from each temperature to find how far each one is from the average.
Each difference is then squared, which ensures deviations are counted positively, and these squared values are averaged to find the variance.

• Differences from mean: [23 - 8.875, 14 - 8.875, 6 - 8.875, -7 - 8.875, -2 - 8.875, 11 - 8.875, 16 - 8.875]
• Squared differences: [198.015625, 26.015625, 8.265625, 253.515625, 118.515625, 4.515625, 50.765625]
• Variance: (sum of squared differences) / 8 = 82.28

A larger variance indicates greater spread among temperatures, while a smaller variance points to values clustering closely around the mean.

Standard deviation complements the variance by translating the spread back into the original units of data, making it more interpretable. While variance is in squared units, the standard deviation provides a clearer indication of spread in terms of the data unit, which is degrees Fahrenheit here. To find standard deviation, we take the square root of the variance:
- Standard deviation = \( \sqrt{82.28} \) = 9.07
This implies that, on average, the temperatures deviate by about 9 degrees from the mean of the data set. It’s a crucial measure in statistics, helping to easily understand dispersion.

Analyzing temperature data involves several statistical measures that give us insights into the data set’s behavior. The calculations of range, variance, and standard deviation allow us to answer important questions, like how varied temperatures are during different days.
- **Range** finds the extent between the coldest and warmest days. - **Variance** unveils how each day's temperature is spread out around the average. - **Standard deviation** gives a clearer, more unit-friendly measure of this spread.
Through these metrics, we discern that the temperatures vary significantly over the eight-day period. Understanding these concepts is critical to making informed decisions based on this temperature data. Whether for predicting trends or planning around future weather events, these statistical tools are invaluable.