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Basic numeracy, statistics
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Basic Numeracy
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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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