Saturday, December 15, 2018

Statistics

Statistics



Find the mean number of plants per house. Statistics deals with collection, presentation, analysis and interpretation of numerical data.

Arranging data in a order to study their salient features is called presentation of data.

Range of the data is the difference between the maximum and the minimum values of the observations

Table that shows the frequency of different values in the given data is called a frequency distribution table

A frequency distribution table that shows the frequency of each individual value in the given data is called an ungrouped frequency distribution table.

A table that shows the frequency of groups of values in the given data is called a grouped frequency distribution table

The groupings used to group the values in given data are called classes or class-intervals. The number of values that each class contains is called the class size or class width. The lower value in a class is called the lower class limit. The higher value in a class is called the upper class limit.

Class mark of a class is the mid value of the two limits of that class.

Mean of Grouped Data
The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations

Mean x̄ 
=  (f1x1 + f1x1 + L + fnxn
  f1 + f1 + L + fn 

Mean x̄ 
= ∑ fixi/fi
where i varies from 1 to n

We can form ungrouped data into grouped data by forming class-intervals of some width.

It is assumed that the frequency of each classinterval is centred around its mid-point.

Class mark 
= (Upper class limit + Lower class limit)/2

Direct method 
Mean x̄ 
= ∑ fixi/fi

Assumed Mean method 
Mean x̄ 
= a + [∑ fidi]/fi

We can only locate a class with the maximum frequency, called the modal class. The mode is a value inside the modal class, and is given by the formula:
Mode
= l + [(f1 - f0)/((2f1) - f0 - f2)]*h

where l = lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal),
f1 = frequency of the modal class,
f0 = frequency of the class preceding the modal class,
f2 = frequency of the class succeeding the modal class.

for finding the median of ungrouped data, we first arrange the data values of the observations in ascending order. Then, if n is odd, the median is the (n+1)/2 th observation. And, if n is even, then the median will be the average of the n/2 th and (n/2)+1 th observations

Less than cumulative frequency distribution:
It is obtained by adding successively the frequencies of all the previous classes including the class against which it is written. The cumulate is started from the lowest to the highest size.

More than cumulative frequency distribution:
It is obtained by finding the cumulate total of frequencies starting from the highest to the lowest class.

To find median class, we find the cumulative frequencies of all the classes and n/2. We now locate the class whose cumulative frequency is greater than (and nearest to) n/2.

After finding the median class, we use the following formula for calculating the median.

Median 
= l + [((n/2) - cf)/f]*h

where 
l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal).

There is a empirical relationship between the three measures of central tendency :
3 Median = Mode + 2 Mean

Representing a cumulative frequency distribution graphically as a cumulative frequency curve, or an ogive of the less than type and of the more than type.

The median of grouped data can be obtained graphically as the x-coordinate of the point of intersection of the two ogives for this data.

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