Measures of Central Tendancy

Measures of Central Tendency are ways of describing the central position of a frequency distribution for a group of data using the mean, median, or mode.  Which measure of central tendency is used will depend on how much and the type of data that was collected collected.

​Type of Data for Responding Variable	​Best Measure of Central Tendency
Qualitative Nominal (no ranking or order)	Mode
​Qualitative Ordinal (has ranking or order)	Median
​Quantitative  With majority of skew values less than |2.0|	Mean
​Quantitative With majority of skew values more than |2.0|	Median

Mean

The arithmetic mean is the most commonly used measure of central tendency.  The mean is used for normally distributed (non-skewed) data.   The mean has one main disadvantage: it is particularly susceptible to the influence of outliers.  These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. 

• DO NOT calculate a mean for skewed data. 
• ​DO NOT calculate a mean from values that are already averages. 
• DO NOT calculate a mean when the measurement scale is not linear (i.e. pH units are not measured on a linear scale)

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The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if we have 'n' number of measurement samples in a data set and they have values x1, x2, … ,xn, the sample mean, usually denoted by x̄ (pronounced "x bar"), is:

Calculating the Mean in Spreadsheet Software

• Excel or Google Sheets = AVERAGE(highlight data cells) highlight the cells with data you want to include in the mean calculation.
• ​Be sure your mean has the correct number of digits. Your processed data should have the same number of digits as your raw data.

Calculating the Mean of a pH

The mean cannot simply be calculated from pH measurements directly because the pH scale is logarithmic; instead, each pH value must be first converted to its corresponding hydrogen ion concentration, then the mean can be calculated from those concentrations.  That average can then be converted back to a pH.

1. Convert pH values to hydrogen ion concentrations:  For each pH value, use the formula [H+] = 10^(-pH) to calculate the corresponding hydrogen ion concentration. 
2. Calculate the average concentration: Add up all the calculated hydrogen ion concentrations and divide by the number of values to find the average concentration. 
3. Convert back to pH: Use the pH formula again: pH = -log(average hydrogen ion concentration) to get the mean pH value.

Median

The median is the middle score for a set of data that has been arranged in order of magnitude. The median should be used if there are outliers or skewed data (meaning the data does not fit a “normal distribution”).  ​ ​

Calculating the Median in Spreadsheet Software

• Online calculators
• ​Excel or Google Sheets = MEDIAN(highlight data cells) highlight the cells with data you want to include in the mean calculation.
• ​Be sure your mean has the correct number of digits. Your processed data should have the same number of digits as your raw data.

Mode

The mode is the most frequent score in a data set.  Calculate the mode by tallying the result that is repeated more often than any other. The mode is the peak of a distribution curve.  Sometimes a distribution as two peaks, which is described as "bimodal."

Calculating the Mode in Spreadsheet Software

• Online calculators​​
  
• ​Excel or Google Sheets = MODE(highlight data cells) highlight the cells with data you want to include in the mean calculation.
• ​Be sure your mean has the correct number of digits. Your processed data should have the same number of digits as your raw data.