Ten Day History of the Weather in South Beach, Statistics Problem Example

The topic which will be reviewed is a ten day history of the weather in South Beach daytime temperature for the ten day period which extends from February 15th to February 25th, 2014. The temperature for a ten day period which range from February 15^th, 2014 to February 25^th, 2014 will be reviewed in this statistics project. The ten daily temperature values from this period will be applied in order to formulate a sample mean, range, sample variance and the sample standard deviation will be initially calculated and the work will be demonstrated (Anderson et al., 2011). A frequency distribution chart, histogram and a graph which demonstrates the distribution of data will be demonstrated. The data which had been collected was accessed from the Friendly Forecast (2014) website. South Beach is located in the southern section of the city of Miami, County of Dade, Florida.

The South Beach community in Miami Beach is a popular tourist destination. South Beach is renowned for its nightlife and recreational activities. The temperature is the south beach community is temperate. The average temperatures in February are 75?. These temperatures make South Beach an  entertaining place to enjoy a break from the wintry weather which has severely plagued the nation during the Winter of 2013- 2014. The moderate temperatures enable tourists to enjoy going to the beach in February (Visit Florida, 2014).

South Beach is also recognized for its international cuisine and Art Deco style of building architecture.  The activities which can be enjoyed in the South Beach community are bicycling, roller blading, tennis, golf and fishing. The capacity of enjoying these recreational activities is attributed to the Caribbean climate of South Beach. The temperature values which were recorded were: 70? F on February 15^th, 71 ? F on February 16^th, 74? F on February 17^th, 80? F on February 18^th, 80?F on February 19^th 81?F on February 20^th, 82?F on February 21^st, 83?F on February 22^nd, 82? F on February 23^rd, 81? F on February 24^th and 81? F on Tuesday February 25^th (Friendly Forecast, 2014). These were temperature values for the weather at South Beach during the period from February 15^th, 2014 to February 25^th, 2014.

The initial chart which will be demonstrated is a frequency distribution. The frequency distribution chart is demonstrated:

Median

In order to find the median value, the temperature values are placed in a list which is composed of the highest to the lowest temperature values.  In this situation the values would be listed as 83? F, 82?F, 81? F, 80? F, 74? F, 71? F and 70? F. The median is derived by extracting the middle value from the list of temperatures for South Beach during the period from February 15^th, 2014 to February 25^th, 2014 (Anderson et al., 2011). The middle value which represents the median of the collection of temperatures is 80? F.

Sample Mean

The sample mean is calculated by applying the following formula:

X_{mean =}1/n (?X_{1- 10})

Where n is the number of values in the collection of south beach temperatures, and the each of the X values represents the daily temperatures. In substituting the value:

1/ 10 (70 + 71+ 74+ 80+ 80+ 80 + 81 + 82 + 83 + 82+ 81 + 81) = 86.5 ?F (Anderson et al., 2011). The sample mean for the temperature values of South Beach for February 15^th to February 25^th, 2014 is 86.5 ?F.

Range

The range is delineated as the differences between the highest value in a collection and the lowest value in a collection (Anderson et al., 2011). In this collection the highest value is 83? F. The lowest value is 70 ? F. The range would be demonstrated as 83- 70. The range is calculated to be 13? F.

Variance

The mean is applied in order to calculate the sample variance. In order to obtain the sample variance, the mean is subtracted from each value and squares. The sample variance is derived from the aggregate of the square of the differences and the mean value which is derived. In this case the sample variance would be calculated by applying the following formula:

Sample variance =1/ n (?X_mean  – X_{1- 10})².

Substituting the temperature values which were derived for South beach during the period from February 15^th to February 25^th 2014, the values are input into the formula are the following;

1/10 [ (80- 70)² + (80 – 71)² +(80 – 74)²+ (80 – 80)² +(80 – 80)²+(80 – 81)² + (80 – 82)² + (80 – 83)²+ (80 – 82)² + (80 – 81)² + (80 – 81)²]. The sample variance for the collection of the south beach temperatures from February 15^th to February 25^th, 2014 is equal to 23.7 ?F (Anderson et al., 2011).

Sample Standard Deviation

The sample standard deviation is derived by taking the square root of the variance. The formula which is applied in order to calculate the sample standard deviation is demonstrated as:

?= [? (X_mean– X_{1- 10})²]^1/2

Where ? is equivalent to the standard deviation, X_mean is the mean value of the temperatures and X_{1- 10} represents the values of each of the ten temperatures which were derived for the South Beach temperatures from February 15^th to February 25^th, 2014. In this case the value of the sample variance is assessed to be 23.7. The sample standard deviation is (23.7)^1/2.  The sample standard deviation is 4.868 ? F (Anderson et al., 2011).

Distribution of Data

As a rule, the first standard deviation is equivalent to 68% of the information from the population sample. The second standard deviation would include 95% of the population derived from the information of the population sample. The third standard deviation would contain 99.7% of the information which is derived from the population sample.  Consequently, the first standard deviation is equivalent to (X_mean – ?). The second standard deviation would be equivalent to (X_mean – 2?) (Anderson et al., 2011).

The third standard deviation would be equivalent to (X_mean – 3?). The values for the first standard deviation would be calculated as (80 – 4.868) = 75.132? F. This standard deviation represents the value in which 685 of the information from the population sample would fall. The second standard deviation would be equivalent to (80- 9.736) = 70.264 ? F. The third and final standard deviation would be equivalent to (80- 13.604) = 66.396 ? F. This standard deviation would represent 99.7% of the information from the population (Anderson et al., 2011).

Conclusion

The values of the temperatures for a ten day period which extends from February 15^th to February 25^th, 2014 were analyzed for the South Beach community in Miami Beach, FL. The frequency distribution, histogram, median, sample mean, range, sample variance, sample standard deviation was reviewed. The aspect of the distribution of data with regards to the first second and third standard deviation was reviewed in this project.

Reference

Anderson, D. R., Sweeney, D. J. & Williams, T.A. (2011). Fundamentals of business     statistics. Belmont, CA: Cengage Learning.

Friendly Forecast (2014). South Beach, Fl. Hourly weather data for February 15^th –        February 25^th, 2014. Friendly Forecast.

Visit Florida (20140. Visitor services South Beach. Visit Florida.

Waner, S. & Costenoble, S. R. (1999). Histogram generator finite mathematic applied to the real world. Hofstra University.