A random sample of 12 joggers was asked to keep track and report the number of miles they ran last week the responses are:
5.5 7.2 1.6 22.0 8.7 2.8
5.3 3.4 12.5 18.6 8.3 6.6
a) Compare the three statistics that measure central location.
b) Briefly describe what each statistic tells you.
calculations
To compare the three statistics that measure central location (also known as measures of central tendency) for the given data set, we need to calculate the mean, median, and mode.
a) Calculation of the three statistics:
Mean: The mean is calculated by summing up all the values and dividing by the number of values in the data set.
Mean = (5.5 + 7.2 + 1.6 + 22.0 + 8.7 + 2.8 + 5.3 + 3.4 + 12.5 + 18.6 + 8.3 + 6.6) / 12 = 6.8667 (rounded to four decimal places)
Median: The median is the middle value when the data set is arranged in ascending or descending order. If there's an even number of values, the median is the average of the two middle values.
Arranging the data in ascending order: 1.6, 2.8, 3.4, 5.3, 5.5, 6.6, 7.2, 8.3, 8.7, 12.5, 18.6, 22.0.
Median = (5.5 + 6.6) / 2 = 6.05
Mode: The mode is the value(s) that occur the most frequently in the data set.
In this case, there are no repeated values, so there is no mode.
b) Interpretation of each statistic:
- Mean: The mean represents the average distance run by the joggers. It is obtained by summing up all the distances and dividing by the number of joggers. In this case, the mean is roughly 6.87 miles, so it indicates that on average, the joggers covered approximately 6.87 miles last week.
- Median: The median is the middle value in the sorted data set. It represents the value that separates the data in half. In this case, the median is 6.05 miles, suggesting that half of the joggers ran more than 6.05 miles and the other half ran less.
- Mode: The mode is the most frequently occurring value in a data set. In this case, since all the values are unique, there is no mode.
To compare the three statistics that measure central location (mean, median, and mode) for the given data set, we need to calculate each of these statistics.
a) Calculations:
1. Mean:
The mean is calculated by summing up all the values in the data set and dividing it by the total number of values.
For the given data set:
Mean = (5.5 + 7.2 + 1.6 + 22.0 + 8.7 + 2.8 + 5.3 + 3.4 + 12.5 + 18.6 + 8.3 + 6.6) / 12
2. Median:
The median is the middle value when the data set is arranged in ascending or descending order. If the data set has an even number of values, the median is the average of the two middle values.
For the given data set, we can arrange the values in ascending order:
1.6, 2.8, 3.4, 5.3, 5.5, 6.6, 7.2, 8.3, 8.7, 12.5, 18.6, 22.0
The median is the middle value, which in this case is the 6th value, so the median = 6.6.
3. Mode:
The mode is the most frequently occurring value in the data set.
For the given data set, there is no value that occurs more than once, so there is no mode.
b) Interpretation:
1. Mean:
The mean provides the average value of the data set. In this case, the mean represents the average number of miles ran by the 12 joggers last week. It takes into account all the values in the data set, which can be influenced by extreme values.
2. Median:
The median is the middle value of the data set. It represents the middle point of the data when arranged in order. In this case, the median represents the "typical" number of miles ran by the joggers, as it is not affected by extreme values.
3. Mode:
The mode represents the most frequently occurring value in the data set. In this case, since there is no value that occurs more than once, there is no mode. A mode would indicate a value that is commonly reported by the joggers.
In summary, the mean provides an average value, the median represents the middle value, and the mode represents the most frequent value in the data set. Each statistic tells us different information about the central location of the data.