Listed below are prices in dollars for one nightat different hotels in a certain region. Find the range , variance, and standard vision for the given sample data. Include appropriate units in the results. How useful are the measures of variation for someone searching for a room. 212 152 192 249 271 190 109 142
To find the range, subtract the lowest value from the highest value:
Highest value = 271
Lowest value = 109
Range = 271 - 109 = 162
Therefore, the range is 162 dollars.
To find the variance, follow these steps:
1. Find the mean (average) of the data:
Total = 212 + 152 + 192 + 249 + 271 + 190 + 109 + 142 = 1507
Mean = Total / Number of values = 1507 / 8 = 188.375
2. Subtract the mean from each value and square the result:
(212 - 188.375)^2 = 546.140625
(152 - 188.375)^2 = 1325.015625
(192 - 188.375)^2 = 13.640625
(249 - 188.375)^2 = 3705.265625
(271 - 188.375)^2 = 6843.265625
(190 - 188.375)^2 = 0.25890625
(109 - 188.375)^2 = 6263.640625
(142 - 188.375)^2 = 2177.890625
3. Find the mean of the squared differences:
Total of squared differences = 546.140625 + 1325.015625 + 13.640625 + 3705.265625 + 6843.265625 + 0.25890625 + 6263.640625 + 2177.890625 = 20875.01875
Mean of squared differences = Total of squared differences / Number of values = 20875.01875 / 8 = 2609.37734375
4. The variance is the mean of the squared differences:
Variance = 2609.37734375
To find the standard deviation, take the square root of the variance:
Standard deviation = √(Variance) = √(2609.37734375)
Therefore, the standard deviation is approximately 51.08568552 dollars.
The range, variance, and standard deviation provide useful information for someone searching for a room. The range gives an idea of the spread of prices, with a difference of 162 dollars between the highest and lowest values. The variance and standard deviation measure the overall variability or dispersion of the prices from the mean. A higher variance or standard deviation would indicate a wider range of prices and potentially more variability in the market. This information can help someone searching for a room to understand the price range and the likelihood of finding rooms at different price points.