. The following data give the 2010 gross domestic product (in billions of dollars) for al 50 states. The data entered in alphabetical order by state (Source: Bureau of Economic Analysis).
173 49 254 103 1901 258 237 62 748 403
67 55 652 276 143 127 163 219 52 295
379 384 270 97 244 36 90 126 60 487
80 1160 425 35 478 148 174 570 49 164
40 255 1207 115 26 424 340 65 248 39
Calculate the mean and median for the data. [16 marks]
Are these values of the mean and median, sample statistics or population parameters? Explain.
For mean, you add up all numbers and divide by the size of the population.
For median, you need to sort them in increasing order, and take the mean of the two middle numbers (average between 25th and 26th values for N=50).
Since the data represent the whole population, they are not sample statistics.
To calculate the mean and median for the given data, first let's arrange the data in ascending order:
26, 35, 36, 39, 40, 49, 49, 52, 55, 60, 62, 65, 67, 80, 90, 97, 103, 115, 116, 127, 143, 148, 163, 164, 173, 174, 190, 237, 244, 248, 254, 255, 258, 270, 276, 295, 340, 379, 384, 403, 425, 424, 458, 487, 570, 652, 748, 1207
To find the mean, add up all the numbers and divide by the total number of values:
Mean = (173 + 49 + 254 + 103 + 1901 + 258 + 237 + 62 + 748 + 403 + 67 + 55 + 652 + 276 + 143 + 127 + 163 + 219 + 52 + 295 + 379 + 384 + 270 + 97 + 244 + 36 +
90 + 126 + 60 + 487 + 80 + 1160 + 425 + 35 + 478 + 148 + 174 + 570 + 49 + 164 + 40 + 255 + 1207 + 115 + 26 + 424) / 50
Mean ≈ 308.72
To find the median, we need to find the middle value of the sorted data. Since we have 50 values, the median will be the average of the 25th and 26th values:
Median = (219 + 52) / 2
Median = 135.5
The mean and median for the given data are approximately 308.72 and 135.5, respectively.
These values of the mean and median are sample statistics because they are calculated from a sample (the given data) and not from the entire population. In order to calculate population parameters, data from the entire population would be needed.
To calculate the mean and median for the given data, follow these steps:
1. Add up all the values in the data set:
173 + 49 + 254 + 103 + 1901 + 258 + 237 + 62 + 748 + 403 + 67 + 55 + 652 + 276 + 143 + 127 + 163 + 219 + 52 + 295 + 379 + 384 + 270 + 97 + 244 + 36 + 90 + 126 + 60 + 487 + 80 + 1160 + 425 + 35 + 478 + 148 + 174 + 570 + 49 + 164 + 40 + 255 + 1207 + 115 + 26 + 424 + 340 + 65 + 248 + 39 = 16,296
2. Count the number of observations in the data set. In this case, there are 50 observations.
3. Calculate the mean by dividing the sum of all the values by the number of observations: Mean = Sum/Number of Observations = 16,296/50 = 325.92
4. To find the median, first arrange the data in ascending order:
26, 35, 39, 40, 49, 49, 52, 55, 60, 62, 65, 67, 80, 90, 97, 103, 115, 1207, 127, 143, 148, 163, 164, 173, 174, 1901, 219, 237, 244, 248, 254, 255, 258, 270, 276, 295, 340, 36, 424, 425, 378, 384, 403, 570, 652, 748, and 1160.
5. Since there are 50 observations, the median will be the middle value. In this case, the middle value is the (50 + 1)/2 = 25.5th value. Since it is not an integer, we need to find the average of the 25th and 26th value. The 25th value is 174, and the 26th value is 1901. Therefore, the median is (174 + 1901)/2 = 1037.5.
Now, let's determine whether the mean and median values are sample statistics or population parameters:
- Mean: The mean calculated in this case is a sample statistic. It represents the average GDP of the 50 states based on the given data set.
- Median: The median calculated in this case is also a sample statistic. It represents the middle value of the GDP data in the sample.
In general, sample statistics are calculated using a subset of the population, while population parameters represent the entire population. Since the given data set represents a subset (sample) of the GDP figures for all 50 states, the mean and median calculated based on this subset are sample statistics.