Finding Mode
Finding Mode
In statistics, the mode is the value that occurs most frequently in a data set or a probability
distribution. In some fields, notably education, sample data are often called scores, and the
sample mode is known as the modal score.
Like the statistical mean and median, the mode is a way of capturing important
information about a random variable or a population in a single quantity.
The mode is in general different from the mean and median, and may be very
different for strongly skewed distributions.
The mode is not necessarily unique, since the same maximum frequency may be
attained at different values.
The most ambiguous case occurs in uniform distributions, wherein all values are
equally likely.
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The mode of a discrete probability distribution is the value x at which its probability
mass function takes its maximum value. In other words, it is the value that is most
likely to be sampled.
The mode of a continuous probability distribution is the value x at which its probability
density function attains its maximum value, so, informally speaking, the mode is at the
peak.
As noted above, the mode is not necessarily unique, since the probability mass
function or probability density function may achieve its maximum value at several
points x1, x2, etc.
The above definition tells us that only global maxima are modes. Slightly confusingly,
when a probability density function has multiple local maxima it is common to refer to
all of the local maxima as modes of the distribution. Such a continuous distribution is
called multimodal (as opposed to unimodal).
In symmetric unimodal distributions, such as the normal (or Gaussian) distribution
(the distribution whose density function, when graphed, gives the famous "bell
curve"), the mean (if defined), median and mode all coincide.
For samples, if it is known that they are drawn from a symmetric distribution, the
sample mean can be used as an estimate of the population mode.
Mode of a sample
The mode of a sample is the element that occurs most often in the collection. For
example, the mode of the sample [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] is 6.
Learn More Frequency Test


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Given the list of data [1, 1, 2, 4, 4] the mode is not unique - the dataset may be said to
be bimodal, while a set with more than two modes may be described as multimodal.
For a sample from a continuous distribution, such as [0.935..., 1.211..., 2.430...,
3.668..., 3.874...], the concept is unusable in its raw form, since each value will occur
precisely once.
The usual practice is to discretize the data by assigning frequency values to intervals
of equal distance, as for making a histogram, effectively replacing the values by the
midpoints of the intervals they are assigned to.
The mode is then the value where the histogram reaches its peak. For small or
middle-sized samples the outcome of this procedure is sensitive to the choice of
interval width if chosen too narrow or too wide; typically one should have a sizable
fraction of the data concentrated in a relatively small number of intervals (5 to 10),
while the fraction of the data falling outside these intervals is also sizable.
An alternate approach is kernel density estimation, which essentially blurs point
samples to produce a continuous estimate of the probability density function which
can provide an estimate of the mode.


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