Finding a Weighted Mean from a Frequency Table

A frequency table may show a value once and list how many observations have that value. Each value must count as many times as its frequency when calculating the mean.

Use the weighted-mean formula

Suppose scores 1, 2 and 3 occur with frequencies 2, 3 and 1. Multiply and add: 1×2 + 2×3 + 3×1 = 11. The total number of observations is 2 + 3 + 1 = 6. The mean is 11 ÷ 6, or about 1.83.

Why row averages are wrong

A simple average of the values 1, 2 and 3 is 2, but that treats each row as if it had one observation. The frequency table contains six observations, with value 2 appearing three times, so the weighted mean is lower than 2.

A table setup

• List each value x.
• List its frequency f.
• Calculate x × f for each row.
• Add the frequencies to get total count N.
• Divide Σ(xf) by N and state the context and units.

Check the result

The weighted mean should lie between the smallest and largest values in the table. If all frequencies are equal, it matches the ordinary average of the listed values. Keep fractions until the end to avoid rounding error.

Read the frequency table before calculating

A frequency is the number of observations represented by a row. If a table lists test scores 60, 70, and 80 with frequencies 2, 5, and 3, the data set contains ten scores, not three. The total points are 60(2) + 70(5) + 80(3) = 710, so the mean is 710/10 = 71. A simple average of the row values would be 70 and would incorrectly give each distinct score equal influence, even though 70 appears more often.

Use a product-and-total table

1. Create columns for value x, frequency f, and product xf.
2. Multiply each value by its frequency; do not add a frequency to the value.
3. Add the frequencies to get the number of observations.
4. Add the products and divide their sum by the total frequency.
5. State the mean in the same units as the values, not in squared units.

For grouped intervals, the exact individual values may not be shown. A common estimate uses each interval's midpoint as the representative value, multiplies that midpoint by the interval frequency, and divides by total frequency. This produces an estimated mean because observations within an interval may not all equal the midpoint. Use the method the question specifies and distinguish an estimated mean from an exact one.

Handle percentages and unequal group sizes

When combining group averages, weight each group's mean by the number of people in that group. If 8 students average 90 and 2 students average 70, the combined mean is (8×90 + 2×70)/10 = 86, not the simple average 80. A weighted mean can also use percentage weights, as long as the weights are expressed consistently and total 1 or 100 percent. If percentages are rounded, a small difference from a reported combined figure may result.

Check whether the answer is plausible

• The result must lie between the smallest and largest values with positive frequency.
• A value with a larger frequency should pull the mean more strongly toward itself.
• If all frequencies are identical, the weighted mean equals the ordinary mean of the listed values.
• If frequencies sum to zero, there are no observations and the mean is undefined.
• Keep exact fractions or decimals through the arithmetic, then round as requested.

A useful verification is to expand a small table into a list by repeating each value its frequency number of times and calculate the ordinary mean. That method is impractical for large data sets but clarifies why multiplication by frequency is necessary.

Use frequency as the weight for each value

A frequency table lists values or categories and how often each occurs. To calculate the mean, multiply each value by its frequency, add the products, and divide by the total frequency: x̄ = Σ(xf)/Σf. For values 2, 4, and 6 with frequencies 3, 5, and 2, the weighted sum is 2(3) + 4(5) + 6(2) = 38 and the total count is 10, so the mean is 3.8.

The table’s frequency column already determines the weights; do not give each row equal influence unless the frequencies are equal. A row with frequency 12 represents twelve observations, not one. Check that the total frequency equals the stated sample size before computing.

Find a missing frequency or value

If a table includes an unknown frequency, use the known mean equation. Suppose scores 60 and 80 have frequencies 4 and x, and the mean is 70. Then (60×4 + 80x)/(4+x) = 70. Multiply through: 240 + 80x = 280 + 70x, so x = 4. Substitute to verify the mean is 70. The result should be a nonnegative whole number if it represents a count.

For grouped intervals, an exact mean is unavailable because the individual observations are unknown. A common estimate uses each class midpoint as the representative value: multiply midpoint by class frequency, sum, and divide by total frequency. Clearly call it an estimate. Wider intervals create more uncertainty about how observations are distributed within each class.

Interpret the weighted result

The weighted mean must lie between the smallest and largest values when all frequencies are nonnegative. It tends to be closer to values that occur more often. If a result falls outside that range, check the total frequency, arithmetic, and whether every row was included. For a categorical table, a numerical mean may not be meaningful unless categories have meaningful numeric values.

Frequency and relative-frequency tables produce the same mean when relative frequencies are calculated from the same observations because each frequency is divided by the same total. Use the form that makes arithmetic easier, and ensure rounded percentages do not cause a small discrepancy.

• Multiply each value or class midpoint by its frequency.
• Divide by total frequency, not number of rows.
• Verify the table totals match the data count.
• Use an algebraic mean equation for a missing count or value.
• Describe midpoint-based calculations for grouped intervals as estimates.

Exam takeaway

Multiply every value by its frequency, add the products and divide by total frequency. Frequency counts weight the result; the number of rows does not.