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Middle number of ascending values.

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Mean, Median, and Mode


The arithmetic mean is more commonly known as the average, and is calculated by dividing the sum of a set of values by the count of the number of values.

  1. Calculate the sum of all of the values in your set
  2. Divide the sum by the number or count of values in the set
  3. The quotient of the sum of the values divided by the number of values is the arithmetic mean

The geometric mean is another type of ‘middle value’, and is calculated by multiplying the values of each member of a set together and taking the  \begin{align*}n^{th}\end{align*} root of the product, where  \begin{align*}n\end{align*} is the number of values in the set.

  1. Multiply the value of each member of the set by the next, as in \begin{align*}x_1 \times x_2 \times x_3 \times x_4\end{align*} , etc.
  2. Find the  \begin{align*}n^{th}\end{align*} root of the product of the set values, where  \begin{align*}n\end{align*} is the number of values in the set
  3. The  \begin{align*}n^{th}\end{align*} root of the product of the set values is the geometric mean of the set

harmonic mean is calculated by dividing the number of values in the set by the sum of the inverses of the values in the set.

  1. Count the number of values in your data set. This number becomes your numerator.
  2. Calculate the sum of the inverses of the data values, this sum becomes your denominator.
  3. Divide the numerator by the denominator, the resulting quotient is the harmonic mean of the values

The median is defined as the value representing the “middle number” of a data set that has been ordered by increasing value, meaning that exactly  \begin{align*}\frac{1}{2}\end{align*} of the data is greater than the median, and \begin{align*}\frac{1}{2}\end{align*} is less.
To identify the median:
  1. Organize your set in ascending numerical order and count the values
  2. If there are an odd number of values, the median is the middle number in the series.  If there is an even number of values, the median is the arithmetic mean of the two middle numbers in the series.

The mode is the value(s) in a set that occurs with the greatest frequency. 
To identify the mode:
  1. Organize the set in numerical order (to make it easier to count repeating values) and make note of the frequencies of any repeated values (any values with a frequency greater than 1)
  2. The value(s) occurring with the greatest frequency are the mode(s)
Find the arithmetic mean, the median, and the mode in this data set:
12 18 11 21 19 18 20 12 4 23 21 12 17 12 11 19 10

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