A sample of blood pressure measurements is taken from a data set and those values (mm Hg) are listed below. The values are matched so that subjects each have systolic and diastolic measurements. Find the mean and median for each of the two samples and then compare the two sets of results. Are the measures of center the best statistics to use with these data? What else might be better? Systolic: Diastolic: 136 150 106 154 104 102 155 97 95 122 59 86 61 81 73 60 90 80 67 71 Find the means The mean for systolic is mm Hg and the mean for diastolic is mm Hg (Type integers or decimals rounded to one decimal place as needed.) Find the medians The median for systolic is mm Hg and the median for diastolic is mm Hg (Type integers or decimals rounded to one decimal place as needed.) Compare the results. Choose the corect answer below
The Correct Answer and Explanation is:
To solve this, let’s break down the steps.
Step 1: Organize the Data
Systolic Measurements:
136, 106, 104, 155, 95, 122, 59, 61, 73, 90, 67, 80
Ordered:
59, 61, 67, 73, 80, 90, 104, 106, 122, 136, 155
Diastolic Measurements:
150, 154, 102, 97, 122, 86, 81, 60, 80, 71
Ordered:
60, 71, 80, 80, 81, 86, 97, 102, 122, 154
Step 2: Calculate the Mean
The mean is the sum of the data divided by the number of data points.
Systolic:
Sum = 59 + 61 + 67 + 73 + 80 + 90 + 104 + 106 + 122 + 136 + 155 = 1133
Number of data points = 11
Mean = 113311≈102.1\frac{1133}{11} \approx 102.1111133≈102.1 mm Hg
Diastolic:
Sum = 60 + 71 + 80 + 80 + 81 + 86 + 97 + 102 + 122 + 154 = 973
Number of data points = 10
Mean = 97310=97.3\frac{973}{10} = 97.310973=97.3 mm Hg
Step 3: Find the Median
The median is the middle value of the ordered data. If the number of data points is odd, the median is the middle value. If it is even, the median is the average of the two middle values.
Systolic:
There are 11 data points (odd), so the median is the 6th value in the ordered list.
Median for systolic = 90 mm Hg.
Diastolic:
There are 10 data points (even), so the median is the average of the 5th and 6th values in the ordered list.
Median for diastolic = 81+862=83.5\frac{81 + 86}{2} = 83.5281+86=83.5 mm Hg.
Step 4: Compare the Results
Means:
- Mean for systolic = 102.1 mm Hg
- Mean for diastolic = 97.3 mm Hg
Medians:
- Median for systolic = 90 mm Hg
- Median for diastolic = 83.5 mm Hg
Step 5: Interpretation of Results
- Systolic vs Diastolic Mean: The mean systolic value (102.1) is higher than the mean diastolic value (97.3), but not by a huge margin.
- Systolic vs Diastolic Median: The median systolic value (90) is also higher than the median diastolic value (83.5).
The mean is affected by extreme values (outliers) in the data. For example, the systolic measurements have a high value of 155, and the diastolic measurements have 154. These high outliers raise the mean values slightly.
The median, however, is not as affected by outliers. It reflects the middle of the data set better. In this case, both the median systolic and diastolic values are slightly closer to each other than the means.
Are the Measures of Center the Best Statistics?
- Mean can be a useful measure, but it can be skewed by extreme values or outliers. In this case, the presence of high outliers (155 for systolic and 154 for diastolic) makes the mean less reliable.
- Median may be a better measure of central tendency in this case because it is less sensitive to extreme values.
Alternative Measure
Another statistic you might consider is the mode, which shows the most frequent values. If any value occurs frequently, it can help show a more common trend.
Conclusion
For this dataset, median might be a better measure of center than the mean due to the presence of outliers in the data. The median gives a more accurate reflection of the typical blood pressure reading.
