Question

Increasing joint extension is one goal of athletic trainers. In a study to investigate the effect of therapy that uses ultrasound and stretching (Trae Tashiro, Masters Thesis, University of Virginia, 2004 ), passive knee extension was measured after treatment. Passive knee extension (in degrees) is given for each of 10 study participants. \(\begin{array}{ll}59 & 46\end{array}\) \(\begin{array}{llll}64 & 49 & 56 & 70\end{array}\) \(\begin{array}{ll}45 & 52\end{array}\) \(\begin{array}{cc}63 & 52\end{array}\) Which would you choose to describe center and spread \(-\) the mean and standard deviation or the median and interquartile range? Justify your choice.

Short answer

Given the dataset isn't skewed or with outliers, and is instead fairly symmetrical, mean and standard deviation would produce a more precise representation of the center and spread of the data compared to median and IQR

Step-by-step solution

Write down the observations

First, write down the observations in a single row: 59, 46, 64, 49, 56, 70, 45, 52, 63, and 52.

Sort the observations

Next, arrange these observations in increasing order: 45, 46, 49, 52, 52, 56, 59, 63, 64, and 70.

Calculate mean, median, standard deviation, and IQR

Compute the average (mean) of the values, which gives a measure of central tendency. Similarly, compute the spread by calculating standard deviation. For median (middle value) and interquartile range (IQR, a measure of statistical dispersion or spread), first find the median. As the number of observations is even, median is the average of 5th and 6th values, which are 52 and 56. Then, split the data into two halves - lower half (values below the median) and upper half (values above the median). The 'median' of the lower half gives first quartile (Q1) and of the upper half gives third quartile (Q3). IQR is then calculated as difference between Q3 and Q1, giving another measure of spread.

Analyze data characteristics

Look for any skewness or outliers, because both mean and standard deviation are influenced by extreme values, whereas median and IQR aren't. Observations do not seem skewed or have any outliers, as there's no value too large or too small.

Decide the preferred measures

As there's no skewness or outliers present, either mean and standard deviation or median and IQR could be used to describe this dataset. However, since the data is pretty symmetrical with mean close to median, mean and standard deviation might give a more precise representation of the data.