SOP: Computation of Frequency Distribution with Median, Mode, Range and Grouped Standard Deviation

GIFSQ — Global Institute for Food Safety & Quality

Principle

Large sets of numerical data — for example, the weights of individual fruits in a lot — are condensed by arranging them in groups (class intervals) and counting how many observations fall in each group. The relation between the class intervals and their frequencies is the frequency distribution, which shows the pattern of the data. The mean, median and mode describe its central tendency, and the range and standard deviation describe its dispersion.

Materials

• The set of individual observations
• A tally sheet for the frequency table

Procedure

Find the maximum and the minimum values in the data and divide the entire range into a suitable number of class intervals of equal width (any observation must belong to only one class; the midpoint of each class interval should preferably be a whole number)

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Tally the observations into the class intervals and record the frequency f of each class

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Find the midpoint x of each class interval and compute fx and fx² for every class

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Add the frequencies to obtain Σf, add the fx values to obtain Σfx, and add the fx² values to obtain Σfx²

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Compute the arithmetic mean from the frequency distribution using the expression below

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Identify the median: for an odd number of observations it is the middlemost value; for an even number it is the mean of the two middlemost values

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Identify the mode: the class interval that occurs most frequently

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Compute the range and the standard deviation of the grouped data using the expressions below

Calculation

Mean from a frequency distribution:

x̄ = Σfx / Σf

Range:

R = Xmax − Xmin

Standard deviation of grouped data:

s = √{[Σfx² − (Σfx)²/Σf] / Σf}

Approximate relation for skew distributions:

Mean − Mode = 3(Mean − Median)

Where,

• x̄ = arithmetic mean
• f = frequency of a class
• x = midpoint of the class interval
• Σf = total number of observations
• R = range
• Xmax = largest observation
• Xmin = smallest observation
• s = standard deviation

Worked example

Weights of 100 pineapples drawn from a lot of 3,000 fruits, grouped into seven class intervals:

Mean: x̄ = 199.4/100 = 1.994 kg

Median: the class interval 1.81–2.00 is the median class. The book illustrates the even-count convention hypothetically: if a mid-value of 2.7 were added to the seven mid-values (1.3, 1.5, 1.7, 1.9, 2.1, 2.3 and 2.5), the two middlemost values would be 1.9 and 2.1, giving a median of (1.9 + 2.1)/2 = 2.0

Mode: the class interval 2.01–2.20, which holds the largest frequency (37)

Range: R = 2.5 − 1.3 = 1.2 kg

Standard deviation: s = √[(405.08 − 199.4²/100)/100] = √(7.4764/100) = 0.2734 kg

Note: the book computes the grouped standard deviation with the divisor Σf (100) rather than Σf − 1; the printed value 0.2734 recomputes exactly on that basis and is carried as printed.

Precautions

• For a frequency distribution to show any definite pattern, the total number of observations should preferably be more than 100.
• Class intervals should be of equal width for better computation of the mean and standard deviation.
• When the distribution is very unsymmetrical, the median is a better measure of central tendency than the mean, which is unduly affected by a long tail. When the distribution is symmetrical, mean, median and mode coincide.
• Neither the median nor the mode finds application in quality control.

References

Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.

Indian Standards 7200, Part 1 (1974) — cited in the book for the frequency-distribution guidance.