Meaning of SKEWNESS AND KURTOSIS

If the frequency of observations has equal distance from an average are the same, then we say that distribution is summetric, that is mean=median=mode.
If the frequency distribution is not symmetric then we say that distribution is asymmetric or skewed.

SKEWNESS

The skewness is the unsymmetric of a frequency distribution. There are two possibilities for a distribution is skewed that is ‘positively skewed ‘ and ‘negatively skewed ‘.

1) Postively Skewed :

The frequency curve has a longer tail on the right of the mode, it is prove that is a positively skewed distribution. Mean greater than median.
Mean > Median > Mode

2) Negatively Skewed :

The frequency curve has the longer tail on the left of the mode.It is prove that it is in a negatively skewed distribution. That is
Mean < Median < Mode.

Different of co-efficient of skewness :

1) Karl peason’s co-efficient
of skewness

a) Mean – Mode
—————————–
SD

b) 3(Mean – Median )
——————————
SD

[SD= Standard Deviation ]

2) Bowley’s co-efficient of
skewness.

(Q3 + Q1) – (2Median)

         Q3 - Q1

KURTOSIS

It refers to the peakness of a frequency distribution. Using the value of beta two the frequency distribution can be divided into 3.They are,

1) Lepto Kurtic distribution

It is a frequency distribution for which beta two grater than 3.

2) Meso Kurtic distribution /
Normal distribution.

It is a frequency distribution for which beta two equal to 3.

3) Platy Kurtic distribution

It is the frequency distribution for which beta two less than 3.

METHODS or MEASURES OF DISPERSION

Following are the important methods
1) Range
2) Quartile Deviation
3) Mean/Average Deviation
4) Standard Deviation

1) Range

It is the simple method of studying dispersion. It is the difference between largest and smallest values in a particular series.

Range=L-S

L=Largest value
S=Smallest value

Co-efficient of range:

The relative measure corresponding to range, is called co-efficient of range.

Co-efficient of range =
(L-S)/(L+S)

2) Quartile Deviation

The measure of dispersion depending upon the lower and upper quartiles is know as the quartile deviation. Quartile are those value which divided the series into 4 equal parts. Hence we have 3quartile Q1, Q2, Q3.

Quartile Deviation
= (Q3-Q1)/2

Co-efficient of quartile deviation:

The relative measure is called ‘Co-efficient of quartile deviation ‘.

QD = (Q3-Q1)/(Q3+Q1)

3) Mean/Average Deviation

Mean deviation is a measure of dispersion. Sum of absolute deviations from an average divided by the number of item.It is also called average deviation.

Mean deviation =
summation of |D| divided
by sum of frequency

Co-efficient of mean deviation:

The calculation of co-efficient of mean deviation is, mean deviation divided by mean.

Co-efficient of mean deviation =
Mean deviation / Mean

4) Standard Deviation

Standard deviation is the measure of an absolute dispersion standard deviation concept was introduced by Karl Pearson in1893.Standard deviation denoted by lower case Greek sigma or SD.

Co-efficient of standard deviation :

Co-efficient standard division is calculated by, standard deviation divided by mean, multiplied by 100.

Co-efficient of standard deviation =
SD / Mean * 100

KINDS OR TYPES OF AVERAGES

There are many type of
averages.Among them mean, median and mode are very important.These are called simple averages.And the other averages are geometric mean and harmonic mean. It is known as special averages.

1)Mean
Simply the mean of a veriable is defined as the sum of the observations divided by the number of observations.This is also
known as arithmetic maen.

Example (1) :
3, 4, 5, 6, 7
Mean = 25/5 = 5
Mean of the series is 5.

2)Median
Median is defined as the middle most observation, when they are arranged in
ascending or descending order.

Example (2):
12, 18, 10, 23, 42, 20, 32
10, 12, 18, [20], 23, 32, 40
(Write the ascending or descending order)
20 is the median.

Example (3):
5, 8, 2, 3, 4, 7
2, 3,[ 4, 5], 7, 8
4+5/2=4.5
Median =4.5

3)Mode
The mode refers to that value is a distribution, which accure most
frequently.

Example (4):
2, 7, [10], 15, 8, [10], 17, 18
Here the mode is 10.

Example (5):
2,[ 7], [10], 8,[ 7], 9, [10]
Here the mode is 10and 7

TYPE OF DATA SERIES IN STATISTICS

There are three type of series. They are individual series, discrete series and continuous series.

1)Individual series
Variables will be given
individually without
grouping class and
frequency.
eg: 2, 4, 6, 8, 10

2)Discrete series
In case of discrete series,
frequency against each of
of the observation is
multiplied by the value of t
the observation. The value,
so obtained are summed
up and divided by the total
number of frequency.
eg: x – 64, 63, 62, 61, 60
f – 8 , 18, 12, 9 , 7

3)Continuous series
In this case, calculating
arithmatic mean is same
as that of discrete series.
The only different is that
the midpoint of various
class intervals are taken.
eg: x – 0-10, 10-20, 20-30
f – 20 , 42 , 55

Kinds of class intervals in
continuous series :

a)Exclusive class interval
0-10, 10-20, 20-30,…..

b)Inclusive class interval
0-9, 10-19, 20-29,…..

c)Unequal class interval
0-20, 20-50, 50-70,….

Characteristics of a good measure of central tendency

An average is a single value that represents a group of value. It lies somewhere between largest and smallest item. For this reason, an average is frequently referred to as a measure of central tendency. It is also known as average or measure of location.

DEFINITION

Simpson and Kafka defined it as “A measure of central tendency is a typical value around which other figures congregate”.

Waugh has expressed “An average stand for the whole group of which it forms a part yet represents the whole”.

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QUALITIES OF A GOOD MEASURE OF CENTRAL TENDENCY

• Rigidly defined
• In some cases would considerably affected the
  value of the average.If the
  average is rigidly defined.
  This instability in it’s value
  would be no more, and it
  would always be a definite
  figure.
• Based on all observation
• To ensure that is should
  represent the entire data
  set, it’s value should be
  calculated by taking into
  consideration the entire
  data set.
• Easy to understand and
  calculate
• The value of an average
  should be computed by
  using a simple method
  without reducing it’s
  accuracy and other
  advantages
• Algebraic treatment
• It should have capable of
  farther algebraic treatment
  In other wards, an ideal
  average is one which can
  be used for further
  statistical calculation.
• Mathematical formula.
• Usually it defined in a form
  of mathematical formula.
  there are different formulas
  For finding average of
  different types of
  Problems.

MEANING AND FUNCTIONS OF STATISTICS

Statistics deals with the collections, analysis, interpretation and presentations of numerical data. It is a method of conducting a study about a particular topic.
According to Croxton and Cowden “statistics may be defined as the collection, presentation, analysis and interpretation of numerical data”.

FUNCTIONS OF STATISTICS

1) Helpful for common man
Statistics is to express number in easily understandable language.

2) Formulating policies
Statistics made a problem simple, and the result of this problem used to create policies.

3) Simple presentation
Statistics present the complex data in a simple format in term of percentage, graphs etc…

4) Helps to forecasting
Extrapolation present data aids in forecasting likely changes that can be expected in future.

5) Compare facts
It facilitate comparison of data and identify the interrelations between large sets of data for drowning suitable inferences.