027. Calculating the Variance and Standard Deviation with Grouped Data

If data are grouped into a frequency table, the variance and standard deviation can be calculated as

. (3.16)

And

. (3.17)

Example 3.9. The flight director for P&P requires information regarding the dispersion of the numbers of passengers. Decisions regarding scheduling and the most efficient size of planes to use depend on the fluctuation in the passenger load. If this variation in number of passengers is large, bigger planes may be needed to avoid overcrowding on those days when the passenger load is extensive. The frequency table for P&P appeared as

Table 3.6 – Frequency Distribution for Passengers

Class (passengers)	Frequency (F) (days)	Midpoint (M)	FM	M2	F M2
50 to 59	3	54.5	163.5	2916	5,832
60 to 69	7	64.5	451.5	3969	19,845
70 to 79	18	74.5	1341.0	5184	72,576
80 to 89	12	84.5	1014.0	6561	118,098
90 to 99	8	94.5	756.0	8100	56,700
100 to 109	2	104.5	208.0	9801	39,204
∑	50		3935.0		312,255

Solution. Given that, the mean was calculated in an earlier example 3.6 as

.

Formulas (3.16) and (3.17) give

passengers squared,

passengers.

Interpretation. The flight director can now decide if the planes currently in use can accommodate fluctuations in passengers levels as measured by a standard deviation of 10.8. If not, perhaps larger planes will be used to accommodate any overflow that might otherwise occur on those days with heavy traffic.

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