Friday, March 16, 2012

Part: 2 Statistics (Central tendency, Arithmetic mean, Median, Geometric mean,Harmonic mean, )

Median: Median is the middle most value of some observations. For ungrounded or raw data, if the number of observations, n is odd,
Median = (n+1/2) th observation
If n is even, Median = {(n/2)th observation + ((n/2)+1)th observation}/2

Example: Median of 1, 7, 10, 15, 20, 25, 30
                              = (7+1)/1 or 4th observation = 15
Example: Median of 1, 7, 10, 15, 20, 25, 30, 33
                              =  {(8/2)th obs. + ((8/2)+1)th obs}/2
                              = (15+20)/2
                              = 17.5

Note: To find out median, observations must be arranged in ascending or descending order.
Mode: Mode is the value of distribution for which the frequency is maximum. Simply, mode is the value of the value of the observation which occurs most. Mode of the observation 1, 7, 9, 11, 11, 15, 15, 15, 20, 21, 27 is 15 (15 occurs 3 times).

Dispersion: Measure of variability
The average help to locate the 'center' or a distribution but they do not reveal how the items or the observations are spread out or scattered. This property is commonly referred to as the 'dispersion', 'scatter' or 'variation'. With can describe almost all distributions with reasonable degree of accuracy.

Dispersion is important because different distributions may have exactly the same averages, but substantial difference is variability. Let us look at the following example:
Two students secured the following marks in an examination:
                          Math.       Stat.      Geo.      Eng.      Ave.
Student 1              49            50        55         54        52
Student 2              64            10        90         40        52

Both the students have the same average marks. But the variations from average were quite different for the two students.
There are many ways  to measure dispersion. One for the most common way is 'the standard deviation'. By definition, the standard deviation is the positive square root of the mean square deviations of the observations from their arithmetic mean.
i.e., Standard deviation (σ) = √(∑(X1-μ))/N
Where X1 = i-th element of the population
              μ = mean of all the observations
             N = total number of observations in the population

Probability:  Probability can be defined as a measure of chance, under classical approach. Probability is calculated under the following formula:
P(A) = n(A)/n(S)
Where, P(A): Probability of concurring the event A
            n(A): No. of favorable outcomes
Simply, probability = (Favorable outcomes/ Total outcomes)

Example: A big contains 4 white and 6 red balls. A ball is drawn at random from the bag. What is the probability that it  is red? white?

Sol. Let, A: the even that the ball is red. B: the even that the ball is white.
:. P(A) = n(A)/n(S) = 6/10 = .6
   P(B) = n(B)/n(S) = 4/10 = .4

Central tendency
In statistics, the term central tendency relates to the way in which quantitative data tend to cluster around some value. A measure of central tendency is any of a .Central Tendency: Introduces the mean, median, and mode.
A guide to the mean, median and mode and which of these measures of central tendency you should use for different types of variable and with skewed ...
Measures of central tendency are measures of the location of the middle or the center of a distribution. The definition of "middle" or "center" is purposely left ...
This lesson describes measures of central tendency, such as the mean, median, mode, proportion, and percentage. From Stat Trek's free, online statistics ...
For a complete lesson on central tendency, or mean median mode range,
To obtain a better understanding of what is considered normal, other measures of central tendency such as the median, the trimmed mean, and the trimean may
The best way to reduce a set of data and still retain its information is to summarize it with a single value.
What will happen to the measures of central tendency if we add the same amount to all data values, or multiply each data value by the same amount?
A measure of central tendency is a measure that tells us where the middle of a bunch of data lies. The three most common measures of central tendency are the ..

Arithmetic mean,
In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers ..
The arithmetic mean of a set of data is presented by Math Goodies. Learn mean at your own pace.
Arithmetic Mean - Definition of Arithmetic Mean on Investopedia - A mathematical representation of the typical value of a series of numbers, computed as the
An arithmetic mean is a fancy term for what most people call an "average." When ... says the average of 10 and 20 is 15, they are referring to the arithmetic mean.
The arithmetic mean of a set of values is the quantity commonly called "the" ... the underlying distribution (known as the population mean), the arithmetic mean of ...
Arithmetic Mean. The most commonly used type of average. To find the arithmetic mean of a set of n numbers, add the numbers in the set and ...
Arithmetic and geometric means, Arithmetic-Geometric Means inequality. General case.
The arithmetic mean is what is commonly called the average of the numbers. The value of $A.M.$ is always between the least and the greatest .
The Arithmetic Mean Calculator is probably the easiest way to learn how to calculate arithmetic mean online because all you have to do is simply enter in a set of ...
geometric mean or means and arithmetic mean how to calculate arithmetic mean center frequency bandwidth calculation average of numbers center frequency .

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. A geometric mean
Practical information on how to calculate 'Geometric Means', including its use for shellfish bed and swimming beach classifications and closures.
Geometric Mean - Definition of Geometric Mean on Investopedia - The average of a set of products, the calculation of which is commonly used to determine the ...
and so on. The geometric mean of a list of numbers may be computed using GeometricMean[list] in the ... the geometric mean is related to the arithmetic mean A ...
Free geometric mean calculations online. Find the geometric average mean value of the given numbers. Can be used for calculating or creating new math ...
the geometric mean is relevant any time several quantities multiply together to produce a product. The geometric mean .
Similar right triangles formed by dropping an altitude--explained using an interactive applet, a you tube video and many step by step example problems.
A kind of average. To find the geometric mean of a set of n numbers, multiply the numbers and then take the nth root of the product.
Math lesson covering the geometric mean, from its basic formula and definition to example problems. Learn how to calculate the geometric mean of a set of ...

Harmonic mean
In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when ...The harmonic mean of a list of numbers may be computed in Mathematica using HarmonicMean[list]. The special cases of n=2 and n=3 are therefore given by ..
Arithmetic and Harmonic Means. A fellow travels from city A to city B. For the first hour, he drove at the constant speed of 20 miles per hour. Then he ...Free Harmonic mean calculations online. Find the Harmonics mean of the given numbers. Can be used for calculating or creating new math statistics problems.John Schuyler project risk and economic decision analysis tip of the week.
To find the harmonic mean of a set of n numbers, add the reciprocals of the numbers in the set, divide the sum by n, then take the reciprocal of ...Like arithmetic mean and geometric mean, harmonic mean is also useful mean and can be defined as Harmonic mean is quotient of number of the given values ...
Many courses offered by the School of Computer Science and Engineering use a harmonic mean formula to calculate students' final results given their exam ...The Harmonic Mean. If three numbers are such that by whatever part of itself the largest term exceeds the middle term, and the middle term exceeds the third by ...

In probability theory and statistics, median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, .Explains how to compute the mean, median, mode, and range of a list of numbers.The Median is the "middle number" (in a sorted list of numbers). ... To find the Median, place the numbers you are given in value order and find the middle ...Statistics, mean, average, median, mode, bimodal and measures of central tendencyThe median of a set of data is presented by Math Goodies. Learn about median at your own pace.noting or pertaining to a plane dividing something into two equal parts, especially one dividing an animal into right and left halves. 2. situated in or pertaining to ..An interactive math lesson about the statistical median.Definition of median: One type of average, found by arranging the values in order and then selecting the one in the middle. If the total number of values in the ...The median of a statistical distribution with cumulative distribution function D(x) is the value x such D(x)=1/2 . For a symmetric distribution, it is therefore equal to ...HyperStat Online Contents. Median. Next section: Mode The median is the middle of a distribution: half the scores are above the median and half are below the ...

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 ...The mode of a set of data is presented in this interactive lesson. Learn about mode at Math Goodies.Similarly, for a data set (2, 4, 9, 6, 4, 6, 6, 2, 8, 2) (right histogram), there are two modes: 2 and 6. A distribution with a single mode is said to be unimodal.a manner of acting or doing; method; way: modern modes of transportation. 2. a particular type or form of something: Heat is a mode of motion. 3. a designated ...The mean, median and mode are types of average. ... The table below shows how to calculate the mean, median, mode and range for two sets of data.The mode is simply the number which appears most often. Finding the Mode. To find the mode, or modal value, first put the numbers in order, then count how ...A record label devoted to New Music, recorded and presented with the highest standards. Composer series include Berio, Cage, Feldman, Reynolds, Subotnick, ...Maths handling data exercise - Mode, median, mean and range activity.Statistics, mean, average, median, mode, bimodal and measures of central tendency.

Dispersion: Measure of variability

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