This PPT is useful to B.Sc and M.Sc Botany students of MS University and other universities

2. • A random sample is one where each item of the population has an
equal chance of being included in the sample.
• There should not be any bias in the selection of the sample.
• A random sample will give close estimates of the population.
• Random sampling is a scientific method of getting a sample from
the population.
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3. • In all statistical analysis, the date collected on qualitative or
quantitative characters may not be suitable to draw inference.
• Hence, summarize the raw data into frequency table for
presentation.
• Frequency distribution is a table that organises data into classes.
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4. • As a general rule, the number of classes should never more than
30 and not less than 6.
• However, the number of classes in a frequency distribution are not
fixed.
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5.  The difference between the highest and lowest values of the observations in
a given set of data is called its range.
 The formula to compute range is
 Range = Highest score – Lowest score
 Example: 5, 12, 13, 13, 14, 15, 15, 15, 18, 20
 The range is 20 – 5 = 15.
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6. • If the sample contains large number of observation, the data is arranged into
different classes and the width of such class is called class interval.
• Class intervals are generally equal.
• As a rule of thumb, five to seven classes are used for sample size up to 50.
• Calculate the class interval using the following formula.
• Class interval
• i =
𝐿 −𝑆
𝐶
• Where
 i = class interval
 L = largest value
 S = smallest value
 C = number of classes
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7. • The class limits are the lowest and highest values which are
included in the class.
• For example – in the class 10-20, the lowest value is 10 and the
highest 20.
• It indicates that there can be no values in the class below the
lowest limit of the class and above the upper limit of the class.
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8. • For each class-interval, we require a point
which would serve as the representative of the
whole class.
• It is assumed that the frequency of each class-
interval is centred on its mid-point.
• The mid-points of the class interval are
calculated by the following formula:
Upper class limit + lower class limit
• Mid-point = ----------------------------------------------
2
• Mid-point is denoted by “x”
Class interval Mid-point (x)
10 – 25 10+25 ÷ 2 = 17.5
25 – 40 25+40 ÷ 2 = 32.5
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9. • The number of times a particular observation occurs in the class interval is
called frequency.
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10. • Tally marks are a quick way of keeping track of numbers in groups of five.
• One vertical line is made for each of the first four numbers; the fourth number
is represented by a diagonal line across the previous four.
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11. • When the data are large and that of continuous variables, the observations are made
into groups to form a frequency distribution.
• Following formula is used to find the arithmetic mean of such a grouped frequency
distribution.
• Mean ( X ) =
𝑓𝑥
𝑛
• Where,
• 𝑓= frequency of that class
• 𝑥 = class mid value
• 𝑓𝑥= product obtained by multiplying the frequency with the class mid value.
• 𝑓𝑥 = total of product obtained by multiplying the frequency with class mid value.
• 𝑛 = number of observations
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12. • Standard deviation is employed to measure the values of dispersion of the individually observed values
around the mean value.
• Standard deviation may also help the experimenter to predict whether the sample studied for a variable
is homogenous or heterogenous.
• Standard deviation is calculated by the following formula. For sample with more than 30 variants
• SD =
( 𝑥−𝑥)2 𝑓
𝑛−1
• Where,
• SD = standard deviation of the sample
• Σ = sign for summation
• 𝑥 = mean
• 𝑥 = class mid value
• 𝑓= frequency of class interval
• 𝑛= number of observations
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13. • 𝑥 − 𝑥 = deviation (𝑑)of class mid value from mean, then SD is calculated by
the following formula
• SD =
𝑓𝑑
2
𝑛−1
• Where,
• Σ = sign for summation
• 𝑓= frequency of class interval
• 𝑑 = deviation
• 𝑛= number of observations
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14. • SE of the mean measures the variability of the mean of the sample. The
formula to calculate SE of the mean is
• SE =
𝑆𝐷
𝑛−1
• SD – standard deviation
• 𝑛 = number of observation
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15. • CV is the SD expressed as a percentage of the mean. It is calculated by the
following formula.
• CV =
𝑆𝐷
𝑥
x100
• Where,
• SD = standard deviation
• 𝑥 = mean
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16. • Histogram is a type of graphical representation of the data collected and
organised.
• To draw a histogram, two axis are required. The horizontal axis (x) shows the
value of class intervals.
• The vertical axis (y) records the class frequencies.
• On each class interval a column is drawn that is as high as the frequency
record for that class.
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