Measures of Central Tendency MARE 250 Dr. Jason Turner

Three most important: Mean Median Modequantitative & qualitative Centracidal Tendencies quantitative

Mean – the average X i = measurement in a population μ = mean of a population N = size of the population Σ = summation; in this case it indicates that all X values are to be added together, and the sum divided by the population size N Calculation of a Population mean: (Zar eq. 3.1) Mean Girls Mean –

Mean Streets Mean – the average X i = measurement in a population x = mean of a sample n = size of the sample ∑ = summation Calculation of a Sample mean: (Zar eq. 3.2)

For Example A sample population of juvenile mahi-mahi X i (in mm): 52, 65, 78, 79, 85, 102, 110, 115, 116, 125 ∑ X i = 927 mm n = mm 10 = 92.7 mm Calculating a Sample Mean

The Median First – arrange measurements in order of magnitude – ascending or descending M = median (Zar eq. 3.4) if the sample size n is an odd number.

The Median – How too... If the number of observations is ODD, then the median is the observation exactly in the middle of the ordered list If the number of observations is EVEN then the median is the mean of the two middle observations in the ordered list In both cases, if we let n denote the number of observations, then the median is at position (n + 1)/2 in the ordered list (Zar eq. 3.4) if the sample size n is an odd number.

For Example M = X (n+1)/2 = X (9+1)/2 = X 5 = 102 A sample population of juvenile mahi-mahi X i (in mm): 65, 78, 79, 85, 102, 110, 115, 116, 125 n = 9 Calculating a Median with Odd sample size

Evenflow M = [X (10/2) +X (10/2) +1 ]/2 M = [5 + 6]/2 M = X 5.5 = 93.5 If the sample size n is an even number, then there is no ‘middle’ value, and M is the average of the two middle values X i (in mm): 52, 65, 78, 79, 85, 102, 110, 115, 116, 125 n = 10 Calculating a Median w/ even sample size

In the event of a tie… In this case where the median value is within a tied set of values (a set of equivalent observations), the median is often interpolated to generate a more accurate estimate. To interpolate means to insert something into a set of existing things; interpolation in math refers to estimate a value between existing values in a case where you need a more accurate value than those given.

Calculating a Median w/ even sample size In the event of a tie… In this case, to generate a more accurate estimate, the median is calculated as: Cum. freq. in this case refers to the cumulative frequency (in numbers, not percent) of the previous classes. Zar eq. 3.5

Calculating a Median w/ even sample size In the event of a tie… M = (89.5) + [0.5(14)-5/4] (1) M = 90 X i (in mm): 52, 65, 78, 79, 85, 90, 90, 90, 90, 102, 110, 115, 116, 125 n = 14

The Mode The mode is typically defined as the most frequently occurring measurement in a set of data The mode is useful if the distribution is skewed or bimodal (having two very pronounced values around which data are concentrated) 30 Number of Individuals

Reach out and Touch Faith First – obtain the frequency of occurrence of each value and note the greatest frequency If the greatest frequency is 1 (i.e., no value occurs more than once), then the data set has no mode If the greatest frequency is 2 or greater, then any value that occurs with that greatest frequency is called a mode of the data set

The Mode In the previous example… X i (in mm): 52, 65, 78, 79, 85, 90, 90, 90, 90, 102, 110, 115, 116, 125 n = 14 The mode is 90

You are so totally skewed! The mean is sensitive to extreme (very large or small) observations and the median is not Therefore – you can determine how skewed your data is by looking at the relationship between median and mean Mean is Greater than the Median Mean and Median are Equal Mean is Less Than the Median

Resistance Measures A resistance measure is not sensitive to the influences of a few extreme observations Median – resistant measure of center Mean – not Resistance of Mean can be improved by using – Trimmed Means – a specified percentage of the smallest and largest observations are removed before computing the mean Will do something like this later when exploring the data and evaluating outliers…(their effects upon the mean)

How To on Computer On Minitab: Your data must be in a single column Go to the 'Stat' menu, and select 'Basic stats', then 'Display descriptive stats'. Select your data column in the 'variables' box. The output will generally go to the session window, or if you select 'graphical summary' in the 'graphs' options, it will be given in a separate window. This will give you a number of basic descriptive stats, though not the mode.