Frequency Tables and Weighted Means

What you’ll learn

Calculate a mean when values have different frequencies.

គណនាមធ្យមភាគពេលតម្លៃមានប្រេកង់ខុសគ្នា។

Meaning and use

A frequency records how often a value occurs. A weighted mean multiplies each value by its frequency before dividing by the total frequency. For grouped intervals, using class midpoints usually gives an estimate rather than the exact mean, because individual values are unknown.

ប្រេកង់បង្ហាញចំនួនដងនៃតម្លៃ។ មធ្យមភាគមានទម្ងន់គុណតម្លៃនីមួយៗនឹងប្រេកង់ រួចចែកសរុបប្រេកង់។ បើប្រើចំណុចកណ្ដាលថ្នាក់ក្រុម ចម្លើយជាទូទៅជាការប៉ាន់ស្មាន។

Formulas

Frequency-weighted mean

\[\bar{x}=\frac{\sum f_i x_i}{\sum f_i}\]

Total frequency

\[N=\sum f_i\]

Symbols and units

Symbol	Meaning	Unit or restriction
xᵢ	Value, or a class midpoint for an estimate	Observation unit
fᵢ	Frequency of that value or class	Nonnegative count
N	Total observations	Positive count
x̄	Mean or estimated mean	Observation unit

Conditions and limits

• The total frequency must be positive.
• Use nonoverlapping categories; a midpoint-based grouped mean is labelled an estimate.
• កុំចែកត្រឹមចំនួនប្រភេទតម្លៃ បើប្រេកង់មិនស្មើគ្នា។

Worked examples

Exact frequency mean

Value 2 occurs three times; value 5 occurs twice.

1. Weighted sum \(2\times3+5\times2=16\).
2. Total count \(3+2=5\).
3. Mean \(16/5=3.2\).

3.2

តម្លៃ 2 ត្រូវរាប់បីដង និង 5 ពីរដង។

An estimated grouped mean

Midpoints 5 and 15 have frequencies 2 and 3.

1. Weighted sum \(5\times2+15\times3=55\).
2. Total frequency 5.
3. Estimated mean \(55/5=11\).

Estimated mean 11

ចំណុចកណ្ដាលជំនួសតម្លៃពិតក្នុងថ្នាក់ក្រុម ដូច្នេះគឺការប៉ាន់ស្មាន។

Common mistakes

Instead of	Use	Why
Mean of value 2 with frequency 3 and value 5 with frequency 2 is (2 + 5)/2.	Mean is (2×3 + 5×2)/5 = 3.2.	Values must be weighted by occurrence.
A midpoint-based mean is always exact.	It is normally an estimate.	The individual values inside each interval are not known.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Value 1 has frequency 2; value 4 has frequency 1. Find the mean.
2. Find total frequency for 3, 4, and 5.
3. Why label a grouped midpoint mean as an estimate?

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 2

   (1×2 + 4×1)/(2 + 1) = 6/3.

2. 12

   Add all frequencies.

3. The actual observations can differ from their class midpoints.

   The calculation substitutes representative values.

Quick recap

• Weight each value by frequency.
• Divide by total frequency.
• Label grouped midpoint results as estimates.