Mean and Range of a Data Set

What you’ll learn

Calculate an average and a simple measure of spread.

គណនាមធ្យមភាគ និងវិសាលភាពនៃទិន្នន័យ។

Meaning and use

The arithmetic mean shares the total equally across the observations. The range is the largest value minus the smallest. Include every observation when counting n, including repetitions. A mean can be affected by an extreme value and need not equal any individual observation.

មធ្យមភាគស្មើផលបូកទិន្នន័យចែកចំនួនទិន្នន័យ។ វិសាលភាពស្មើតម្លៃធំបំផុតដកតូចបំផុត។ ត្រូវរាប់តម្លៃដែលស្ទួនដែរ។

Formulas

Arithmetic mean

\[\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}\]

Range

\[R=x_{\max}-x_{\min}\]

Symbols and units

Symbol	Meaning	Unit or restriction
x₁ … xₙ	All observed values	Same unit
n	Number of observations	Count; must be positive
x̄	Arithmetic mean	Same unit as observations
R	Range	Same unit as observations

Conditions and limits

• At least one observation is needed for a mean.
• Use comparable measurements; do not average incompatible units.
• មធ្យមភាគអាចមិនស្មើតម្លៃណាមួយក្នុងទិន្នន័យ។

Worked examples

A four-value mean

Times are 4, 6, 8, and 10 minutes.

1. Total \(4+6+8+10=28\) min.
2. Count \(n=4\).
3. Mean \(\bar{x}=28/4=7\) min.

7 min

បូកតម្លៃទាំងអស់ រួចចែកចំនួនតម្លៃ។

Repeated observations

Scores are 2, 2, 5, and 7. Find mean and range.

1. Mean \((2+2+5+7)/4=4\).
2. Range \(7-2=5\).

Mean 4; range 5

ចំនួន 2 ដែលស្ទួនត្រូវរាប់ពីរដង។

Common mistakes

Instead of	Use	Why
The mean of 2, 2, 5 is (2 + 5)/2.	The mean is (2 + 2 + 5)/3 = 3.	Repeated observations still count.
Range = largest + smallest.	Range = largest − smallest.	Range measures the distance between the extremes.

Practice

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

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Find the mean of 3, 5, and 7.
2. Find the range of 1, 4, 4, and 9.
3. Must the mean appear in the data set?

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 5

   The total is 15 and there are three values.

2. 8

   Largest 9 minus smallest 1.

3. No.

   For 1 and 2, the mean is 1.5, which is not an observed value.

Quick recap

• Count repeated observations.
• Mean and range measure different features.
• Keep the original measurement unit.