basic numeracy-statistics

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Statistics

Statistics

The branch of Mathematics which deals with collection, classification and

interpretation of data is called statistics.

When used in the singular sense, statistics refers to the subject as a whole of

science of statistical methods embodying the theory and techniques. When it is

used in the plural sense, statistics refers to the data itself (ie, numerical facts

collected in a systematic manner with some definite purpose in view, in any field

of enquiry).

The Frequency Table or the Frequency Distribution

If the data is classified in a convenient way and presented in a table it is called

frequency table or frequency distribution.

Frequency: When the data is presented in a frequency table, the number of

observations that fall in any particular class is called the frequency of that class.



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Class Limit: The starting and end values of each class are called “lower limit”

and “upper limit” of that class respectively.

Class-interval: The difference between the upper and lower boundary of a class

is called the “class-interval” or “size of the class”. It can also be defined as the

difference between the lower or upper limits or boundaries of two consecutive

classes.

Class Boundaries: The average of the upper limit of a class and the lower limit

of the succeeding class is called the “upper boundary” of that class. The upper

boundary of a class becomes the “lower boundary” of the next class.

Range: The difference between the highest and the lowest observation of a data

is called its range.

Histogram: Pertaining to a frequency distribution, if the true limits of the classes

are taken on the x-axis and the corresponding frequencies on the y-axis and

adjacent rectangles are drawn, the diagram is called „histogram‟.



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Frequency Polygon and Frequency Curve: If the points pertaining to the mid

values of the classes of a frequency distribution and the corresponding

frequencies are plotted on a graph sheet and these points are joined by straight

lines, the figure formed is called frequency polygon. If these points are joined by a

smooth curve the figure formed is called frequency curve.

Cumulative Frequency Curves: If the points pertaining to the boundaries of the

classes of a frequency distribution and the corresponding cumulative frequencies

are plotted on a graph sheet and they are joined by a smooth curve, the figure

formed is called cumulative frequency curve.

The figure formed with upper boundaries of the classes and the corresponding

less than cumulative frequencies is called less than cumulative frequency curve.

The figure formed with lower boundaries of the classes and the corresponding

greater than cumulative frequencies is called greater than cumulative frequency

curve.



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Arithmetic Mean (AM) or Mean

1. Arithmetic Mean of Ungrouped Data

If x1, x2 , x3 ,... , xn are n values of a variable x, then arithmetic mean x is defined

as

Where = (x1 + x2 + x3 +...+ xn )

2. Arithmetic Mean of Grouped Data

Here, the mean may be computed by the following method

Direct method If x1, x2 , x3 ,... , xn are n values of a variable x and fl, f2, f3,... fn are

the corresponding frequencies, then

Where, = f1x1 + f2x2 +...+ fnxn and N = f1 + f2 + f3 +...+ fn



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Median

1. Median of Ungrouped Data

If x1, x2, ... , xn are n values of variable x arranged in order of increasing or

decreasing magnitude then the middle-most value in this arrangement is called

the median.

If n is odd, then the median will be the ( n+1 / 2 ) value arranged in order of

magnitude. In this case there will be one and only one value of the median.

If n is even, then the data arranged in order of magnitude, will have 2 middle -

most values ie, ( n / 2)th and ( n / 2 +1) values.



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2. Median of Grouped Data

If N is the number of observation, we first calculate N / 2 . Then from the

cumulative frequency distribution, we determine the class in which observation

lies. Let us name this as median class. We use the following formula for

calculating the median.

Median (M) =

Where, l = lower boundary of the median class

ie, the class where the ( N / 2 )th observation lies.

N = total frequency

F = cumulative frequency of a class preceding the median class.

f = frequency of the median class.

C = length of the class interval.



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Mode

The mode or modal value of a distribution is that value of the variable for which

the frequency is maximum. For a given data, mode may or may not exist.

If mode exists for a given data, it may or may not be unique. Data having unique

mode is called uni-modal. While the data having two modes is called bi-modal.

1. Mode of Ungrouped Data

The observation with the highest frequency becomes the mode of the data.

2. Mode of Grouped Data

Mode =

Where, l = lower boundary of the modal class.

f = frequency of the modal class.

f1 = frequency of the class preceding the modal class.

f2 = frequency of the class following the modal class.

C = length of the class interval.



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Relation between Mean, Median and Mode

3(Median) – 2(Mean) = Mode

Measure of Dispersion

The dispersion of data is the measure of spreading (scatter) of the data about

some central tendency. It is measured in the following types.

(a) Range (b) Quartile Deviation

(c) Mean Deviation (d) Standard Deviation

(e) Variance.

1. Range

Range is the difference of maximum and minimum values of the data.

Range = Maximum value - Minimum value.



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2. Quartile Deviation

After arranging the data in the ascending order of magnitude we find

QD =

Where, for even observation

Q1 = ( n/4)th observation and Q3 = ( 3n/4)th observation

for odd observation.

Q1 = (n+1/4)th observation and Q3 = 3(n+1)th / 4 observation.

3. Mean Deviation

It is defined as the arithmetic mean of the absolute deviations of all the values

taken about any central value. The mean deviation of a set of n numbers x1, x2,

... , xn is defined by

MD =



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4. Standard Deviation

The standard deviation of a set of n numbers x1, x2 , ... , xn is defined by

Standard deviation is denoted by the Greek letter (sigma).

5. Variance

The variance of a set of it numbers x1, x2, ... , xn is defined as the square of the

standard deviation.



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6. Coefficient of Variation

It is given by , where is the standard deviation and is the

mean of the given observations.

Example 1: If the numbers 3, 4, 6 and 8 occur with frequencies 1, 5, 2 and 4

respectively, then the arithmetic mean is

Solution.



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Example 2: The weight of 20 students are given below:

Weight (kg) 40 42 45 48

Number of students 6 8 4 2

Find the mean weight.

Weight (kg) xi Frequency fi fixi

40 6 240

42 8 336

45 4 180

48 2 96

Total f = 20 fixi = 852

Solution



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basic numeracy-statistics

Education

frequency distribution

frequency table

class limit

frequency polygon

class boundaries

cumulative frequency

succeeding class

particular class