Most probable value given a dataset

Question

Forgive my high-school level mathematics but say I have the following dataset:

[2, 3, 4, 5, 6, 6.5, 7, 7.5, 8.5, 9.5]

I want to know what value is most likely to occur in this given dataset. I would typically be after the mode value but each value has a frequency of 1. The mean is 5.9, but I don't believe that is the most probable value.

[6, 6.5, 7, 7.5] are the more interesting subset of values as they are closer in value compared to the rest of the set. However there are 4 values below that subset and 2 values above. So logically I would say I'm looking for a value around 6.5

A solution would be to round the values to:

[2, 3, 4, 5, 6, 7, 7, 8, 9, 10] giving a mode of 7 - or 6.75 if I average the original values.

I'm sure there's a mathematical formula or algorithm that I'm unaware of to find the most likely value of a given dataset. What would be the best approach to solving this?

Answer 1

From your question and comments I have impression that you are after mode.

I would typically be after the mode value but each value has a frequency of 1.

It is simply not true that one cannot compute mode for continuous variables

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 has its maximum value, so the mode is at the peak. (Wikipedia)

Below you can see simple function in R that calculates kernel density using data vectors you used as examples and then takes maximum density point.

x <- c(1, 2, 3, 4, 5, 6, 6.75, 6.85, 6.9, 7, 7.1, 7.15, 7.25)
y <- c(2, 3, 4, 5, 6, 6.5, 7, 7.5, 8.5, 9.5)
z <- c(2, 3, 4, 5, 6, 7, 7, 8, 9, 10)

dmode <- function(x, ...) {
  dx <- density(x, ...)
  dx$x[which.max(dx$y)]
} 

> dmode(x)
[1] 6.70214
> dmode(y)
[1] 6.70214
> dmode(z)
[1] 6.9075

Kernel density estimates, modes (red) and individual datapoints plotted as rug on the top side of the plot for x and y are shown below.

Answer 2

The are several techniques in descriptive statistic: you cited the mean, the mode and the median.

You can also say this: assuming that the data you are working with have a normal distribution, you can say that approximately the 68% of the values are between the mean and one standard deviation, the 95% between the mean and two standard deviations and 99% between the mean and three standard deviations.

With your data set the results are:

68% between 3,5 - 8,3