Statistics and Dispersion

A frequency distribution with two modes is called bimodal. Unimodal has one mode, and trimodal has three.

The geometric mean involves taking the nth root of the product of n numbers. If any number is negative, the root may be imaginary; if zero, the product is zero. Thus, it is typically defined for positive observations.

The median is a positional average and is not affected by extreme values (outliers) in the dataset, unlike the arithmetic mean.

Like the geometric mean, the harmonic mean is not suitable for further algebraic treatment because it is based on the reciprocal of the values.

Harmonic mean gives more weight to smaller values because of the reciprocal in its formula. A small number becomes large when reciprocated, dominating the calculation.

mean $=\dfrac{\sum {x _1}}{n}=\dfrac{90}{11}\approx 8.18$

If a constant is subtracted from every observation in a set, the mean of the set is also decreased by that same constant.

The mode is the value that appears most frequently. In the set 10, 19, 22, 16, 15, 18, 20, 18, 14, 18, 23, the number 18 appears three times, which is more than any other number.

The second quartile (Q2) is the median. First, sort the data: 10, 14, 15, 16, 18, 18, 18, 19, 20, 22, 23. With 11 observations, the median is the 6th value, which is 18.

The median (Md) divides the data into two halves (50%), the second quartile (Q2) is the median, the 5th decile (D5) is the median, and the 50th percentile (P50) is the median. Thus, Md = Q2 = D5 = P50.

The arithmetic mean is the sum of observations divided by the count. (10 + (-10)) / 2 = 0 / 2 = 0.

The sum of the first n natural numbers is n(n+1)/2. The average is the sum divided by n, which results in (n+1)/2.

By definition, the mode of a frequency distribution is the value that occurs most frequently, meaning it has the maximum frequency.

For a series with N observations, if N is odd, the median is the ((N+1)/2)th observation. Substituting N = 2n+1 gives ((2n+1+1)/2)th = (2n+2)/2 = (n+1)th observation.

In a moderately skewed distribution, the empirical relationship is: Mean - Mode = 3(Mean - Median). Given Mean = 10 and Mode = 7, we have 10 - 7 = 3(10 - Median), so 3 = 3(10 - Median), which means 1 = 10 - Median, so Median = 9.