Question

Find the range, the sample variance and the sample standard deviation. In a psychology experiment, 10 subjects were given 5 minutes to complete a task. Their time on task (in seconds) is recorded. $$ \begin{array}{lllll} 175 & 190 & 250 & 230 & 240 \\ 200 & 185 & 190 & 225 & 265 \end{array} $$

Short answer

Answer: The range is 90 seconds, the sample variance is 961.67, and the sample standard deviation is 31.02 seconds.

Step-by-step solution

Find the range

To find the range, we first identify the highest and lowest values in the dataset. Upon examination, we see that the lowest value is 175 and the highest value is 265. The range is calculated as: Range = Highest value - Lowest value Range = 265 - 175 = 90 seconds

Calculate the mean of the dataset

To calculate the mean, we will sum the data values and divide by the number of subjects (10). Mean = (175 + 190 + 250 + 230 + 240 + 200 + 185 + 190 + 225 + 265) / 10 Mean = 2150 / 10 = 215 seconds

Compute the sample variance

To calculate the sample variance, we first compute the squared differences from the mean for each subject and then find the sum of those squared differences. Finally, we divide the sum by the number of subjects minus 1 (since it's a sample). Sample variance = sum of squared differences / (number of subjects - 1) = ((175-215)^2 +(190-215)^2 + (250-215)^2 + (230-215)^2 + (240-215)^2 +(200-215)^2 + (185-215)^2 + (190-215)^2 +(225-215)^2 +(265-215)^2) /(10-1) = (1600 + 625 + 1225 + 225 + 625 + 225 + 900 + 625 + 100 + 2500) / 9 = 8650 / 9 = 961.67 (rounded to two decimal places)

Calculate the sample standard deviation

To find the sample standard deviation, we simply take the square root of the sample variance found in the previous step: Sample standard deviation = √961.67 = 31.02 seconds (rounded to two decimal places) The range is 90 seconds, the sample variance is 961.67, and the sample standard deviation is 31.02 seconds.

Key concepts

Understanding Range in Descriptive Statistics

In descriptive statistics, the range is a measure of how spread apart the values in a dataset are. It's the difference between the largest and smallest values. In our psychology experiment, where subjects completed a task in varying amounts of time, finding the range is straightforward: we subtract the lowest time, 175 seconds, from the highest time, 265 seconds, which gives us a range of 90 seconds.

This tells us that there's a 90-second spread from the fastest to the slowest completion times. A large range can indicate that the data points are spread out over a wide set of values, while a small range indicates that they are more closely clustered. However, the range doesn't take into account any of the other values in the set, and a single outlier can greatly impact it, which is why other measures of spread, such as variance and standard deviation, are also important.

Sample Variance Explained

The sample variance is a statistic that tells us how much the values in a sample differ from the mean of the sample. It is calculated by taking the sum of the squares of the differences between each data point and the mean, and then dividing that sum by the number of observations minus one. The 'minus one' part is crucial because it corrects the bias in the estimation of the population variance from a sample.

In our example with task completion times, the sample variance is 961.67 after rounding to two decimal places. This high variance suggests that there is considerable variability in how long different subjects took to complete the task. Knowing the variance is helpful, but it's in squared units. It's often more intuitive to look at the standard deviation, which is the square root of the variance and in the same units as the data, to understand variability.

Sample Standard Deviation as a Measure of Spread

Whereas the sample variance provides a squared estimate of variability, the sample standard deviation gives us that estimate in the original units of the data, making it easier to interpret. It tells us, on average, how much individual data points deviate from the mean of the dataset.

In the task completion experiment, the sample standard deviation is 31.02 seconds, rounded to two decimal places. This value means that, on average, the subjects' completion times were around 31 seconds away from the mean time. When comparing datasets or assessing the reliability of the data, the standard deviation serves as a useful yardstick. A smaller standard deviation implies more consistency among data points, while a larger one indicates greater levels of discrepancy.