As part of studying the relationship between mental and physical health, you
obtain the following heart rates:
73 72 67 74 78 84 79 71 76 76
79 81 75 80 78 76 78
In terms of differences in heart rates, interpret these data using the following:
(a) the range, (b) the variance, and (c) the standard deviation.
(a) The range of heart rates is like a roller coaster ride—sometimes up, sometimes down, but always thrilling. In this case, the range would be calculated as the difference between the highest and lowest heart rate. So, buckle up and prepare for a heart-pounding calculation!
The highest heart rate in this dataset is 84, while the lowest is 67. Now, let's subtract the lowest heart rate from the highest heart rate: 84 - 67 = 17. So, the range of these heart rates is 17 beats per minute. It's like the difference between riding a snail and a cheetah—quite a surprising range!
(b) Oh, variance, why do you always have to complicate things? Variance is like trying to make sense of a clown car with too many clowns—just when you think you've got it figured out, more clowns pop out!
The variance is a measure of how spread out the heart rates are. To calculate the variance, we need to find the average heart rate first. Adding up all the heart rates, we get a total of 1,254. Since we have 17 heart rates, let's divide the total by 17: 1,254 / 17 = 73.7647 (approximately).
Now, it's time to calculate the variance by subtracting each heart rate from the average, squaring the result, and then adding them all up. But let me tell you, it's going to be a wild ride of calculations. I'm so glad I can do it in a blink as a bot!
After doing all the calculations, we find that the variance is approximately 20.23. It's like trying to juggle pies while riding a unicycle—a bit tricky!
(c) Hold on tight, because now we're diving into the world of standard deviation. Standard deviation is like the bumpy ride of a pothole-filled road—it measures how much the heart rates deviate from the mean, and there's bound to be some bumps in the road!
To calculate the standard deviation, we take the square root of the variance we calculated earlier. So, the square root of approximately 20.23 turns out to be around 4.50 (rounded to two decimal places). It's like trying to balance on a tightrope while juggling rubber chickens—a delicate balancing act!
To interpret the data in terms of the range, variance, and standard deviation, we first need to calculate these measures. Let's go step by step:
(a) The range: The range is the difference between the highest and lowest values in a dataset. To calculate the range, we need to find the maximum and minimum values in the heart rate data.
The maximum value in the dataset is 84, and the minimum value is 67. Therefore, the range is 84 - 67 = 17.
(b) The variance: Variance is a measure of how spread out the values in a dataset are. To calculate the variance, we need to find the average of the squared differences between each heart rate and the mean.
1. Find the mean (average) heart rate:
To calculate the mean, add up all the heart rates and divide by the total number of heart rates:
73 + 72 + 67 + 74 + 78 + 84 + 79 + 71 + 76 + 76 + 79 + 81 + 75 + 80 + 78 + 76 + 78 = 1313
Mean = 1313 / 17 = 77.24 (rounded to two decimal places)
2. Calculate the squared differences from the mean for each value:
For each heart rate, subtract the mean from it and square the result. Then sum up all these squared differences.
Squared differences:
(73 - 77.24)^2 + (72 - 77.24)^2 + (67 - 77.24)^2 + (74 - 77.24)^2 + (78 - 77.24)^2 + (84 - 77.24)^2 + (79 - 77.24)^2 + (71 - 77.24)^2 + (76 - 77.24)^2 + (76 - 77.24)^2 + (79 - 77.24)^2 + (81 - 77.24)^2 + (75 - 77.24)^2 + (80 - 77.24)^2 + (78 - 77.24)^2 + (76 - 77.24)^2 + (78 - 77.24)^2
Sum of squared differences = 369.59 (rounded to two decimal places)
3. Calculate the variance:
Variance = sum of squared differences / (number of heart rates - 1)
Variance = 369.59 / (17 - 1) = 23.10 (rounded to two decimal places)
(c) The standard deviation: The standard deviation is the square root of the variance. It measures the average amount of variation or dispersion in the dataset.
Standard deviation = sqrt(variance) = sqrt(23.10) ≈ 4.81 (rounded to two decimal places)
Now, interpreting these measures in relation to the heart rate data:
(a) The range is 17. This means that the difference between the highest and lowest heart rates in the dataset is 17.
(b) The variance is 23.10. This indicates that there is a considerable amount of variability or spread among the heart rates in the dataset.
(c) The standard deviation is 4.81. This tells us that, on average, the heart rates in the dataset deviate from the mean by approximately 4.81 units. A higher standard deviation suggests greater variability in the data.
By calculating and interpreting the range, variance, and standard deviation, we can gain insights into the dispersion and variation of the heart rates in the dataset.