What Is the Mean? Definition and Formula
The arithmetic mean is the sum of all values in a dataset divided by the number of values. It is the most common measure of central tendency and represents the "typical" or "average" value in a numerical dataset.
x̄ = (Σx) / n | μ = (ΣX) / N
In the formula above, x̄ (pronounced "x-bar") is the sample mean and μ (mu) is the population mean. The symbol Σx means "the sum of all x values." The letter n refers to the number of observations in a sample, while N refers to the total number of individuals in a full population.
x̄ = sample mean
Σx = sum of all values
n = number of observations
μ = population mean (uses N)
The mean is part of descriptive statistics — the branch of statistics concerned with summarizing data. Alongside the median and mode, the mean is one of the three primary measures of central tendency. Understanding when to use each measure is covered in detail in the decision guide further below.
The MEAN Framework: 4 Steps to Calculate Any Mean
Before working through the examples, it helps to have a consistent procedure. The MEAN Framework below applies to every dataset in this guide — from five student scores to 12 months of rainfall data.
🔢 THE MEAN FRAMEWORK
Every worked example in this guide follows the MEAN Framework so you can recognize the same four-step pattern across very different domains — from sports scoring averages to hospital patient records.
Mean Examples in Education
Education is where most students first encounter the arithmetic mean. Teachers compute mean test scores to track class performance; universities compute GPA using weighted means. The examples below cover both standard and weighted calculations.
Example 1 — Mean Test Score (Class of 8 Students)
Problem: Eight students in a statistics class scored the following marks on a quiz: 72, 85, 90, 68, 95, 78, 82, 61. What is the class mean score?
Measure: n = 8 students. Data type: numerical (test scores). No missing values.
Evaluate: Scores range from 61 to 95. No extreme outliers. Sample mean (x̄) is appropriate.
Add: Σx = 72 + 85 + 90 + 68 + 95 + 78 + 82 + 61 = 631
Normalize: x̄ = 631 / 8 = 78.875 ≈ 78.9
✅ The class mean quiz score is 78.9 out of 100. This tells the teacher that the "typical" student scored near a C+/B– boundary. Five of the eight students scored above the mean, and three scored below it.
Example 2 — Weighted Mean GPA
Problem: A student completes four university courses. Grades and credit hours are: Statistics (A = 4.0, 3 credits), English (B = 3.0, 3 credits), Physics (C+ = 2.3, 4 credits), and History (A– = 3.7, 2 credits). What is the student's GPA?
When observations do not carry equal importance, a weighted mean is used. GPA uses credit hours as weights because a 4-credit course contributes more to a student's average than a 2-credit course.
Multiply each grade by its credit weight:
Statistics: 4.0 × 3 = 12.0
English: 3.0 × 3 = 9.0
Physics: 2.3 × 4 = 9.2
History: 3.7 × 2 = 7.4
Sum the weighted values: Σ(w·x) = 12.0 + 9.0 + 9.2 + 7.4 = 37.6
Sum the weights (total credits): Σw = 3 + 3 + 4 + 2 = 12
Weighted Mean GPA: x̄_w = 37.6 / 12 = 3.133
✅ The student's semester GPA is 3.13 (B equivalent). The Physics grade (2.3, worth 4 credits) pulled the average below what a simple unweighted average would show. Unweighted mean = (4.0 + 3.0 + 2.3 + 3.7) / 4 = 3.25, which overstates the student's performance relative to credit load.
Example 3 — Mean Daily Attendance
Problem: A school records daily student attendance for one week: Monday 312, Tuesday 298, Wednesday 305, Thursday 287, Friday 271. What is the mean daily attendance?
Sum: 312 + 298 + 305 + 287 + 271 = 1,473
Divide: x̄ = 1,473 / 5 = 294.6
✅ Mean daily attendance is 294.6 students. Friday's 271-student figure is below the mean, which administrators might flag as a recurring attendance issue worth investigating.
Mean Examples in Healthcare
Clinical researchers use the mean to summarize patient data — blood pressure readings, drug response times, recovery durations, and laboratory measurements. The mean appears in almost every published clinical study as a primary summary statistic.
Example 4 — Mean Blood Pressure (Patient Group)
Problem: A clinical trial records systolic blood pressure (mmHg) for 10 patients before treatment: 138, 145, 128, 155, 141, 133, 162, 127, 149, 140. What is the mean blood pressure?
Sum: 138 + 145 + 128 + 155 + 141 + 133 + 162 + 127 + 149 + 140 = 1,418
Divide: x̄ = 1,418 / 10 = 141.8 mmHg
✅ Mean systolic BP = 141.8 mmHg. The American Heart Association classifies this as Stage 2 Hypertension (≥ 140 mmHg). The researcher uses this mean as a baseline; after treatment, a new mean is calculated and compared using a paired t-test.
Example 5 — Mean Hospital Length of Stay
Problem: A hospital's orthopedic ward records length of stay (in days) for 8 knee-replacement patients: 3, 4, 3, 5, 4, 7, 3, 4. What is the mean length of stay?
Evaluate: The value 7 is higher than the rest. It could represent a complication case. The mean will be slightly higher than the typical 3–4 day stay.
Sum: 3 + 4 + 3 + 5 + 4 + 7 + 3 + 4 = 33
Divide: x̄ = 33 / 8 = 4.125 days
✅ Mean length of stay = 4.1 days. The median is 4.0 days. The one outlier (7 days) nudged the mean slightly above the median. Hospital administrators use this figure to estimate bed availability and resource planning.
Mean Examples in Business and Finance
Businesses use the arithmetic mean daily — to report average revenue, benchmark employee compensation, track average transaction values, and forecast demand. Finance professionals use the mean to calculate expected returns and measure portfolio performance.
Example 6 — Mean Employee Salary (Outlier Effect)
Problem: A tech startup has 6 employees. Their annual salaries are: $48,000, $52,000, $55,000, $51,000, $49,000, and $350,000 (the founder/CEO). What is the mean salary? Is it representative?
Sum: $48,000 + $52,000 + $55,000 + $51,000 + $49,000 + $350,000 = $605,000
Divide: x̄ = $605,000 / 6 = $100,833
⚠️ Mean salary = $100,833 — but five of six employees earn roughly $51,000. The CEO's $350,000 salary is an outlier that inflates the mean by ~$50,000 above the typical wage. The median salary ($51,500) is more representative here. This is a textbook case of why outliers matter when interpreting the mean.
When income, price, or asset data contains extreme high values, the mean is pulled upward and can misrepresent the "typical" case. The U.S. Census Bureau and most economic reporting agencies use median household income, not mean income, for this reason. Read more about handling outliers in descriptive statistics.
Example 7 — Mean Daily Sales Revenue
Problem: A coffee shop records daily revenue for one week (Mon–Sun): $840, $920, $780, $1,050, $1,180, $1,460, $1,290. What is the mean daily revenue?
Sum: 840 + 920 + 780 + 1,050 + 1,180 + 1,460 + 1,290 = $7,520
Divide: x̄ = $7,520 / 7 = $1,074.29
✅ Mean daily revenue = $1,074. The owner can project approximate weekly revenue as: 7 × $1,074 = $7,518. Weekday revenue ($780–$920) is consistently below the mean; weekend revenue ($1,180–$1,460) is consistently above it. This pattern suggests staffing and inventory decisions should differ by day of week.
Example 8 — Mean Monthly Stock Return
Problem: An investment portfolio records monthly returns (%) for six months: 2.1, −1.4, 3.8, 0.9, −0.6, 4.2. What is the mean monthly return?
Sum: 2.1 + (−1.4) + 3.8 + 0.9 + (−0.6) + 4.2 = 9.0
Divide: x̄ = 9.0 / 6 = 1.5% per month
✅ Mean monthly return = 1.5%. Annualized simple return estimate: 1.5% × 12 = 18% per year. Note: For compound returns over time, the geometric mean is more accurate than the arithmetic mean. The arithmetic mean is appropriate here because it represents the expected return in any given future month.
Mean Examples in Weather and Climate
Meteorologists rely on the mean to define "normal" climate conditions. Monthly mean temperature, annual mean rainfall, and mean wind speed are all computed using the arithmetic mean over large datasets. These benchmarks help forecasters identify anomalies and trends.
Example 9 — Mean Daily Temperature
Problem: A city records daily high temperatures (°C) for one week in July: 31, 33, 29, 35, 38, 36, 32. What is the mean high temperature for the week?
Sum: 31 + 33 + 29 + 35 + 38 + 36 + 32 = 234
Divide: x̄ = 234 / 7 = 33.43°C
✅ Mean daily high = 33.4°C (92°F). This figure is compared against the 30-year historical monthly mean from climate records to determine whether the week was warmer or cooler than normal. The World Meteorological Organization uses 30-year means as "climate normals."
Example 10 — Mean Monthly Rainfall
Problem: Monthly rainfall (mm) for a region over 12 months: 45, 38, 62, 90, 115, 145, 180, 162, 98, 55, 42, 30. What is the mean monthly rainfall?
Sum: 45 + 38 + 62 + 90 + 115 + 145 + 180 + 162 + 98 + 55 + 42 + 30 = 1,062 mm
Divide: x̄ = 1,062 / 12 = 88.5 mm per month
✅ Mean monthly rainfall = 88.5 mm. Total annual rainfall = 1,062 mm. The monsoon months (July–August) are far above the mean; January–February are far below it. The mean alone does not capture this seasonal variation — a histogram would show the full pattern.
Mean Examples in Sports Analytics
Sports analytics departments in professional leagues use arithmetic means to evaluate player performance, compare team statistics across seasons, and set benchmarks for scouting decisions. The examples below reflect real-world applications in basketball, cricket, and athletics.
Example 11 — Mean Points Per Game (Basketball)
Problem: A basketball player scores the following points in 10 games: 22, 18, 31, 25, 19, 28, 14, 33, 27, 23. What is the player's mean points per game (PPG)?
Sum: 22 + 18 + 31 + 25 + 19 + 28 + 14 + 33 + 27 + 23 = 240
Divide: x̄ = 240 / 10 = 24.0 PPG
✅ Mean points per game = 24.0. This is an All-Star-level average. The player's low game (14 points) and high game (33 points) reveal a 19-point range — the standard deviation would quantify this game-to-game variability.
Example 12 — Mean Race Finishing Time
Problem: Six marathon runners from a local club finish with times (in minutes): 242, 258, 271, 235, 264, 249. What is their mean finishing time?
Sum: 242 + 258 + 271 + 235 + 264 + 249 = 1,519
Divide: x̄ = 1,519 / 6 = 253.2 minutes (≈ 4 hrs 13 min)
✅ Mean finishing time = 253.2 minutes (4 hours, 13 minutes, 12 seconds). The coach uses this figure to set a team training benchmark and identify runners who are significantly above or below the group average.
Mean Examples in Manufacturing and Quality Control
In manufacturing, the arithmetic mean is central to statistical process control (SPC). Engineers track the mean of measurements — weight, diameter, tensile strength — to verify that production stays within tolerance limits. A drift in the mean signals a process problem before defects accumulate.
Example 13 — Mean Bottle Fill Weight
Problem: A bottling plant samples 10 bottles. Their fill weights (grams) are: 500.1, 499.8, 500.4, 499.6, 500.3, 500.0, 499.9, 500.2, 499.7, 500.5. The target weight is 500g. What is the mean fill weight?
Sum: 500.1 + 499.8 + 500.4 + 499.6 + 500.3 + 500.0 + 499.9 + 500.2 + 499.7 + 500.5 = 5,000.5
Divide: x̄ = 5,000.5 / 10 = 500.05 g
✅ Mean fill weight = 500.05 g. This is only 0.05g above the 500g target — the process is well-controlled. Quality engineers would also compute the standard deviation to confirm the spread is within ±3σ control limits, following the NIST Statistical Process Control guidelines.
Mean Examples in Everyday Life
The arithmetic mean appears in daily life far more than most people realize. Fuel efficiency, electricity bills, commute times, household budgeting — all rely on mean calculations to make routine decisions.
Example 14 — Mean Commute Time
Problem: A commuter tracks their travel time to work (minutes) over 5 days: 38, 42, 35, 51, 44. What is the mean commute time?
Sum: 38 + 42 + 35 + 51 + 44 = 210
Divide: x̄ = 210 / 5 = 42.0 minutes
✅ Mean commute = 42 minutes. Planning to leave 50 minutes before any meeting covers the mean plus some buffer for the occasional delay (like Tuesday's 51-minute commute).
Example 15 — Mean Monthly Household Expense
Problem: A family's monthly grocery bills for 6 months: $420, $390, $465, $410, $480, $355. What is their mean monthly grocery spend?
Sum: 420 + 390 + 465 + 410 + 480 + 355 = $2,520
Divide: x̄ = $2,520 / 6 = $420 per month
✅ Mean grocery spend = $420/month. The family can budget $420 for grocery costs. December's $355 low-spend may reflect holiday travel (fewer meals at home), so they should revisit the mean after the full year to get a more representative figure.
Example 16 — Mean Fuel Efficiency (MPG)
Problem: A driver records fuel economy (miles per gallon) over 8 fill-ups: 32.4, 31.8, 33.1, 30.9, 32.7, 31.5, 34.0, 32.2. What is the mean MPG?
Sum: 32.4 + 31.8 + 33.1 + 30.9 + 32.7 + 31.5 + 34.0 + 32.2 = 258.6
Divide: x̄ = 258.6 / 8 = 32.33 MPG
✅ Mean fuel economy = 32.3 MPG. This is the best number to use when estimating how far the car will travel on a full tank or planning road trip fuel costs.
Mean Examples in Agriculture and Science
Example 17 — Mean Crop Yield Per Acre
Problem: A farmer measures wheat yield (kg) from 7 different fields: 3,450, 3,620, 3,290, 3,780, 3,510, 3,660, 3,400. What is the mean yield per field?
Sum: 3,450 + 3,620 + 3,290 + 3,780 + 3,510 + 3,660 + 3,400 = 24,710 kg
Divide: x̄ = 24,710 / 7 = 3,530 kg per field
✅ Mean yield = 3,530 kg/field. The lowest-performing field (3,290 kg) is 240 kg below the mean — the farmer may investigate whether soil, irrigation, or seed variety differences explain the shortfall.
Quick-Reference: 13 Additional Mean Examples
The table below provides 13 more real-life mean examples in compact format. Each shows the dataset, the sum (Σx), the count (n), and the resulting mean (x̄).
| Domain | Dataset (abbreviated) | Σx | n | x̄ (Mean) |
|---|---|---|---|---|
| Website Visitors / Day | 1,240; 1,380; 950; 1,620; 1,100; 1,490; 1,310 | 9,090 | 7 | 1,298.6 visitors |
| Product Ratings (e-commerce) | 4, 5, 3, 5, 4, 2, 5, 4, 3, 5 | 40 | 10 | 4.0 / 5 stars |
| Student Sleep (hours/night) | 6.5, 7.0, 5.5, 8.0, 6.0, 7.5, 6.5 | 47.0 | 7 | 6.71 hours |
| Call Center Handle Time (min) | 4.2, 6.8, 3.1, 5.5, 7.2, 4.9, 5.1 | 36.8 | 7 | 5.26 min |
| House Prices in Suburb ($K) | 320, 415, 280, 510, 375, 440, 390 | 2,730 | 7 | $390K |
| Infant Birth Weights (kg) | 3.1, 2.9, 3.4, 3.2, 2.8, 3.6, 3.0 | 22.0 | 7 | 3.14 kg |
| Chemical Reaction Time (sec) | 48, 52, 45, 55, 50, 47, 53 | 350 | 7 | 50.0 seconds |
| App Session Duration (min) | 3.2, 8.5, 2.1, 12.3, 4.7, 6.8, 5.4 | 43.0 | 7 | 6.14 min |
| Electricity Bill ($/month) | $112, $98, $134, $121, $108, $145, $117 | $835 | 7 | $119.29 |
| Patient Recovery Days | 5, 7, 4, 9, 6, 5, 8 | 44 | 7 | 6.29 days |
| Exam Anxiety Score (1–10) | 6, 8, 5, 9, 7, 4, 7 | 46 | 7 | 6.57 / 10 |
| Retail Foot Traffic / Hour | 42, 68, 55, 71, 60, 49, 83 | 428 | 7 | 61.1 customers |
| Carbon Emissions (tonnes/yr) | 2.4, 3.1, 1.8, 2.9, 3.5, 2.2, 2.7 | 18.6 | 7 | 2.66 tonnes |
Mean vs Median vs Mode: Complete Comparison
The three most common measures of central tendency — mean, median, and mode — answer the same basic question ("what is the typical value?") in different ways. Choosing the right one depends on the shape of your data, whether outliers are present, and what you want to communicate.
| Feature | Mean (x̄) | Median | Mode |
|---|---|---|---|
| Definition | Sum ÷ count | Middle value when sorted | Most frequently occurring value |
| Symbol | x̄ or μ | M or Md | Mo |
| Affected by outliers? | Yes — strongly | No — resistant | No — resistant |
| Data type required | Numerical (interval/ratio) | Numerical or ordinal | Any (including categorical) |
| Unique? | Always (one value) | Always (one value) | May be multiple modes |
| Uses all data points? | Yes | No (uses only middle values) | No (uses only frequent values) |
| Ideal distribution | Symmetric, no outliers | Skewed, with outliers | Categorical, bimodal data |
| Example use case | Mean exam score | Median household income | Most popular shoe size |
| Formula | x̄ = Σx / n | Middle of sorted list | Most frequent value |
| Related page | Mean Guide | Median Guide | Mode Guide |
For a more detailed exploration of when each measure is appropriate, see Mean vs Median vs Mode: When to Use Each on the Statistics Fundamentals blog. For population vs sample comparisons specifically, see the population vs sample guide.
Types of Mean: Arithmetic, Weighted, Geometric, and Harmonic
The arithmetic mean is the most common type, but statistics recognizes three additional mean types used in specific contexts.
| Type of Mean | Formula | When to Use | Real-Life Example |
|---|---|---|---|
| Arithmetic Mean | x̄ = Σx / n | Equally weighted numerical data without extreme skew | Average test score, mean temperature |
| Weighted Mean | x̄_w = Σ(w·x) / Σw | When data points carry different levels of importance (weights) | GPA calculation, index funds, survey responses |
| Geometric Mean | G = (x₁ · x₂ · … · xₙ)^(1/n) | Rates, ratios, and compounding growth figures | Mean investment growth rate, average inflation rate |
| Harmonic Mean | H = n / Σ(1/xᵢ) | Rates and ratios where the denominator varies | Average speed over equal distances, P/E ratio analysis |
Decision Guide: When to Use the Mean
📊 Should You Use the Mean, Median, or Mode?
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⚡ Arithmetic Mean Calculator
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How to Calculate the Mean in Excel, R, and Python
For datasets larger than what you can sum by hand, software tools handle the arithmetic instantly. Below are the exact commands for the four most common environments used by students and analysts.
| Tool | Function / Command | Example |
|---|---|---|
| Microsoft Excel | =AVERAGE(range) | =AVERAGE(A1:A20) |
| Google Sheets | =AVERAGE(range) | =AVERAGE(B2:B15) |
| Python (NumPy) | numpy.mean(array) | np.mean([72, 85, 90, 68]) |
| Python (pandas) | df['col'].mean() | df['scores'].mean() |
| R | mean(vector) | mean(c(72, 85, 90, 68)) |
| SQL (PostgreSQL) | AVG(column) | SELECT AVG(score) FROM results; |
Common Misconceptions About the Mean
| What People Believe | Why It's Misleading | The Correct Understanding |
|---|---|---|
| "The mean always represents the typical value" | Not true when outliers are present | With outliers, the median is a better measure of the "typical" value |
| "Mean and average always mean the same thing" | In statistics, "average" can mean mean, median, or mode | In everyday usage they're interchangeable; in statistics, be precise |
| "A higher mean is always better" | Depends entirely on context | Mean errors should be minimized; mean revenue should be maximized — direction matters |
| "The mean must be one of the values in the dataset" | Incorrect — the mean can be any value | Mean of {2, 3} = 2.5, which is not in the dataset. This is expected and valid |
| "You can always use the mean for numerical data" | Not when data is ordinal with unequal spacing | Likert scale data (1–5 ratings) is ordinal; many statisticians prefer median for such data |
| "The mean of means equals the overall mean" | Only true when group sizes are equal | With unequal groups, use a weighted mean: overall x̄ = Σ(nᵢ · x̄ᵢ) / Σnᵢ |
Entity and Formula Glossary
| Term | Symbol / Formula | Definition |
|---|---|---|
| Arithmetic Mean | x̄ = Σx / n | Sum of all values divided by the number of observations; the standard "average." |
| Population Mean | μ = ΣX / N | Arithmetic mean of every individual in the complete population. |
| Sample Mean | x̄ = Σx / n | Arithmetic mean of a subset drawn from the population; used to estimate μ. |
| Weighted Mean | x̄_w = Σ(wᵢxᵢ)/Σwᵢ | Mean where each value is multiplied by a weight reflecting its relative importance. |
| Geometric Mean | G = (x₁x₂…xₙ)^(1/n) | nth root of the product of all values; used for rates and growth figures. |
| Harmonic Mean | H = n / Σ(1/xᵢ) | Reciprocal of the mean of reciprocals; used when averaging rates. |
| Median | M | Middle value of a sorted dataset; resistant to outliers. |
| Mode | Mo | Most frequently occurring value; applicable to any data type. |
| Dataset | {x₁, x₂, …, xₙ} | A collection of measured observations for a defined variable. |
| Observation | xᵢ | A single recorded data point within a dataset. |
| Outlier | — | A value that lies far from most other values, distorting the mean. |
| Central Tendency | — | A single representative value describing the centre of a distribution. |
| Standard Deviation | s or σ | Average distance of each data point from the mean; measures spread. |
| Variance | s² or σ² | Squared average distance from the mean; the square of standard deviation. |
| Range | max − min | Difference between the largest and smallest values in a dataset. |
| Descriptive Statistics | — | Branch of statistics that summarizes and describes data using measures like mean, median, and standard deviation. |
| Frequency Distribution | — | A table showing how often each value (or range of values) appears in a dataset. |
| Histogram | — | Bar chart showing the frequency distribution of numerical data grouped into intervals. |
Frequently Asked Questions
What is the mean with an example?
The mean is the sum of all values divided by the count of values. Example: test scores of 72, 85, 90, 68, 95. Sum = 410. n = 5. Mean = 410 / 5 = 82. Interpretation: the student's typical test performance is 82 out of 100.
What is the difference between mean and median?
The mean uses every value in the calculation, making it sensitive to outliers. The median is the middle value when data is sorted, so extreme values don't affect it. For symmetric data, mean ≈ median. For skewed data or data with outliers, they can differ significantly. See the full comparison in Mean vs Median vs Mode.
Why is the mean important in statistics?
The mean is the foundation of almost every advanced statistical calculation. The standard deviation measures spread around the mean. Confidence intervals are built around sample means. Hypothesis tests compare means between groups. Without the mean, most of inferential statistics collapses.
Can the mean be negative?
Yes. If the dataset contains negative values (e.g., temperatures below zero, financial losses, negative returns), the mean can be negative. Example: portfolio monthly returns of −2%, −5%, +1%, −3% give a mean of −2.25% per month.
How does adding a constant affect the mean?
Adding a constant c to every value in the dataset increases the mean by exactly c. Multiplying every value by a constant k multiplies the mean by k. These properties make the mean useful in z-score standardization, where you subtract the mean and divide by the standard deviation to shift and scale the distribution.
What is the mean used for in data science?
Data scientists use the mean for feature engineering, imputing missing values, normalizing data (mean centering), evaluating model performance (Mean Absolute Error, Mean Squared Error), and describing dataset summaries in exploratory data analysis (EDA). The pandas DataFrame .mean() method and numpy.mean() are among the most-called functions in data science workflows.
What is the mean used for in healthcare research?
Clinical researchers report mean outcome measures (e.g., mean blood pressure, mean recovery time, mean lab values) in every randomized controlled trial. The statistical difference between the mean of a treatment group and a control group is assessed with a two-sample t-test or paired t-test.
Sources and Further Reading
The examples, formulas, and classifications in this guide follow established statistical references cited below. For academic or professional use, consult the primary sources directly.
- NIST/SEMATECH e-Handbook of Statistical Methods — Chapter 1.3.5 – Measures of Location
- OpenStax Statistics — Chapter 2.5: Measures of the Center of the Data
- Khan Academy — Mean, Median, and Mode Review
- American Statistical Association — K–12 Statistics Education Resources
- NumPy Documentation — numpy.mean() reference
- Microsoft Excel Documentation — AVERAGE Function
- Fisher, R.A. (1925) — Statistical Methods for Research Workers. Edinburgh: Oliver and Boyd. (Foundational work establishing sample mean methodology.)
- For hypothesis testing of means, visit Hypothesis Testing Examples on Statistics Fundamentals.