When we collect or are given a set of numbers which represent multiple observations of a single variable;
for instance, the ages of a group of people, the prices of a set of products or the temperatures measured over a series of
days, we often start by taking a look at the so-called 'univariate' statistics of that variable. Examples of those
statistics are
minimum and maximum,
mean,
median,
mode, and
standard deviation (σ).
Each of these represents a specific property of the collective (whole) set of numbers. Below you can play with these; here
we give you the definitions:
The minimum and maximum express the extreme values of a data set; i.e., the lowest and highest values
respectively.
The mean (or average) is meant to measure the 'center' of the data set. We typically recognize
three different types of mean:
Arithmetic mean (μ):This is the most commonly used mean. It is computed by summing all the numbers in the set and dividing that
sum by the amount of numbers in the set (n):
\[ \mu = \frac{1}{n} \sum_{i=1}^{n} x_i \]
Geometric mean: computed by multiplying all the numbers in the set and by taking the nth root of that
product where n is the amount of numbers in the set:
\[ \left( \prod_{i=1}^{n} x_i \right)^{\frac{1}{n}} \]
Note: the geometric mean cannot be computed for datasets which have one or more negative numbers.
Harmonic mean: computed by dividing the amount of numbers in the set (n) by the sum of the inverses
of all the numbers in the set:
\[ H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} \]
Note: the harmonic mean cannot computed for datasets that contain one or more zeroes as that
would imply division by zero.
These means are related in the following way (unless all the numbers in the set are identical):
Harmonic mean ≤ Geometric mean ≤ Arithmetic mean
The median is even more a measure of the 'center' of a set of numbers as it represents the middle value of the set
if the set would be sorted from low to high or high to low. In case the set has an even number of values,
the median is the mean of the two middle numbers of the sorted set.
The mode of a set of values is the value that occurs the most times. If multiple values occur equal
number of times than all those values are considered part of the mode and the data set is considered
multimodal.
Whereas mean and median are different ways to measure the 'center' of the data,
the standard deviation (σ) measures the 'spread' or variability of the data. It is computed by taking
the square root of the sum of all squared differences between
the values in the set and the arithmetic mean (μ) of the set:
\[ \sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}} \]
You can play with these below. Select numbers from the number table and/or type them in the textbox. Feel free to
pick the same number multiple times. The order of picking numbers does not matter. Once you are ready to see the
statistics, hit the Do it! button.
Your numbers:
(Numbers must be comma-separated, but spaces are optional)