Nestor R. answered • 06/26/19
Professional experienced using mathematics and statistics
First order the concentrations from lowest to highest:
10.41, 11.20, 11.85, 11.95, 12.61, 12.91, 13.01, 13.49, 13.86, 14.54
The median is the 2nd quartile. It is the middle value of the set of data. Since there are n=10 data points, the median is the average of the middle 2 data points. Median = (12.61+12.91)/2 = 12.76.
The 1st quartile Q1 is the value of the n x 0.25th data point. 10 x .25 = 2.5. Since this product is not an integer, the rule is to use the value in the position of the next highest integer. The next highest integer is 3, so Q1 = 11.85.
The same reasoning applies to finding the 3rd quartile Q3. It is the data point in the 8th position; Q3 = 13.49.
So, the quartiles are: Q1=11.85, Q2 (median)=12.76 and Q3=13.49.
The mean X-bar is simply the sum of the data values divided by the number of data points.
X-bar=(10.41+11.20+11.85+ ... +13.86+14.54)/10 = 125.83/10 = 12.583.
The sample standard deviation s is the square root of sum of the squared differences (Xi - X-bar) divided by n-1.
Each data point is a value Xi. Calculate [10.41-12.583)^2 + (11.20-12.583)^2 + ... + (14.54-12.583)^2)] / 9.
Take the square root of this results to get s=1.2537.
The reason you're dividing by 9 instead of 10 when finding the standard deviation is because you're dealing with a sample coming from a presumably larger population. Essentially a sample with mean x-bar and standard deviation s, has n-1 slots available for different data, but the value of the nth slot is forced on you. That's what is referred to as degrees of freedom, and for a sample it is n-1.
