Session 6.1 — Descriptive Statistics

Mean • Median • Mode • Min/Max • Range • Variance • Standard Deviation — definitions + clean equations (no LaTeX) + Excel + live examples

Input numbers Separate with commas, spaces, or new lines.

Use sample formulas? Sample (n−1)

Or try:

Count (n)

—

Mean

—

Median

—

Mode(s)

—

If multiple, all shown.

Min / Max

—

Range: —

Variance

—

Std dev: —

Show work / steps

Sorted data

#	x(i)	x(i) − mean	(x(i) − mean)²
Totals:		—	—

Equations (clean text)

Mean: x̄ = (1/n) Σ xᵢ

Median: odd → x_((n+1)/2); even → ½( x_(n/2) + x_(n/2+1) )

Range: max − min

Variance: population σ² = Σ(xᵢ−μ)² / N; sample s² = Σ(xᵢ−x̄)² / (n−1)

Std dev: σ = √(σ²); s = √(s²)

What each one is • Equation • Excel • Live Example

Mean (Average)

What it is

Balance point of the data; add everything and divide by how many values.

Equation

x̄ = (1/n) Σ xᵢ

Excel

=AVERAGE(A2:A1000)

Example

Mean = —

Median

What it is

Middle value when data are sorted (50% below, 50% above).

Equation

odd → x_((n+1)/2); even → ½( x_(n/2) + x_(n/2+1) )

Excel

=MEDIAN(A2:A1000)

Example

Median = — (—)

Mode

What it is

Most frequent value(s); there can be zero, one, or several modes.

Equation

value(s) with the highest frequency

Excel

=MODE.SNGL(A2:A1000)

=MODE.MULT(A2:A1000)

Example

Mode(s) = —

Min • Max • Range

What it is

Smallest value, largest value, and their difference.

Equation

min xᵢ, max xᵢ; range = max − min

Excel

=MIN(A2:A1000)

=MAX(A2:A1000)

=MAX(A2:A1000)-MIN(A2:A1000)

Example

Min = —, Max = —, Range = —

Variance

What it is

Average squared distance from the mean (spread). Population uses N; sample uses n−1.

Equation

σ² = Σ(xᵢ−μ)² / N; s² = Σ(xᵢ−x̄)² / (n−1)

Excel

=VAR.P(A2:A1000)

=VAR.S(A2:A1000)

Example

s² = — (Σ(x−mean)² = —, divide by —)

Standard Deviation

What it is

Square root of variance; typical distance from the mean, in original units.

Equation

σ = √(σ²); s = √(s²)

Excel

=STDEV.P(A2:A1000)

=STDEV.S(A2:A1000)

Example

s = —

Student Q&A — Descriptive Statistics (6.1)

Q1: Mean vs Median — which should I report?

A: If the distribution is roughly symmetric (no big outliers), report the mean. If it is skewed or has outliers, use the median (and consider IQR).

IQR = QUARTILE.INC(range,3) − QUARTILE.INC(range,1)

Q2: Median rule for odd vs even n?

A: Sort first.

odd n → x_((n+1)/2)

even n → ½( x_(n/2) + x_(n/2+1) )

Excel

=MEDIAN(A2:A1000)

Q3: Sample vs Population stdev/variance — what’s the difference?

A: Population uses N (you have every value). Sample uses n−1 (degrees of freedom) when your data are a subset of a larger group.

Population: σ² = Σ(x−μ)² / N; σ = √(σ²)

Sample: s² = Σ(x−x̄)² / (n−1); s = √(s²)

Excel (stdev)

STDEV.P(range) • STDEV.S(range)

Excel (variance)

VAR.P(range) • VAR.S(range)

Q4: What if there are multiple modes? Or none?

A: There may be zero, one, or several modes. Report all modes or say “no mode”.

Excel

MODE.SNGL(range) • MODE.MULT(range)

Q5: Range vs IQR — which shows spread better?

A: Range = max−min is easy but outlier‑sensitive. IQR = Q3−Q1 is robust, so pair median + IQR for skewed data.

Excel IQR

=QUARTILE.INC(range,3)-QUARTILE.INC(range,1)

Q6: Can standard deviation be negative? What about zero?

A: Stdev is never negative. It is 0 only when all values are identical.

Q7: Best way to summarize for a lab report?

Symmetric data: mean ± stdev
Skewed or with outliers: median (IQR)
Always report n and the units.

Q8: What are σ, μ vs x̄, s?

A: Greek letters (μ, σ, σ²) = population parameters; Latin (x̄, s, s²) = sample statistics.

Excel Data Analysis → Descriptive Statistics (ToolPak)

Use Excel’s built‑in Data Analysis add‑in to generate a one‑click summary table (mean, median, mode, stdev, variance, skewness, kurtosis, min, max, range, sum, count, confidence level).

Enable the ToolPak (one time)

Windows: File → Options → Add‑ins → bottom “Manage: Excel Add‑ins” → Go… → check Analysis ToolPak → OK.
Mac: Tools → Excel Add‑ins… → check Analysis ToolPak → OK. (If prompted, allow installation.)

Run Descriptive Statistics

Go to the Data tab → click Data Analysis (far right).
Choose Descriptive Statistics → OK.
Input Range: e.g., $A$2:$A$1000 (contiguous data).
- Grouped By: Columns (typical). Use Rows only if each row is a separate variable.
- Labels in first row if your range includes a header (e.g., “Heights”).
Output options: pick Output Range (e.g., $D$2) or New Worksheet Ply.
Check Summary statistics. (Optional) Check Confidence Level for Mean and enter 95 for a 95% CI margin of error.
Click OK → Excel prints a table of results.

Note: ToolPak’s “Standard Deviation” and “Sample Variance” are sample formulas (divide by n−1). If you need population versions, use STDEV.P and VAR.P.

What the ToolPak shows → Do it yourself with formulas

Output	DIY Excel formula	Notes
Mean	=AVERAGE(range)
Standard Error	=STDEV.S(range)/SQRT(COUNTA(range))	Uses sample stdev
Median	=MEDIAN(range)
Mode	=MODE.SNGL(range)	Use MODE.MULT for multi‑modal
Standard Deviation	=STDEV.S(range)	Sample version
Sample Variance	=VAR.S(range)	Sample version
Skewness	=SKEW(range)	Sample skewness
Kurtosis	=KURT(range)	Excess kurtosis
Range	=MAX(range)-MIN(range)
Minimum	=MIN(range)
Maximum	=MAX(range)
Sum	=SUM(range)
Count	=COUNTA(range)	Counts non‑blanks
Confidence Level (95%)	=CONFIDENCE.T(0.05, STDEV.S(range), COUNTA(range))	Margin of error; CI ≈ mean ± value

95% CI (full sentence)

=AVERAGE(range) & " ± " & CONFIDENCE.T(0.05, STDEV.S(range), COUNTA(range))

Troubleshooting & tips

No Data Analysis button? The add‑in isn’t enabled; follow the steps above.
Blank or text cells: ToolPak ignores blanks and text. Convert numbers‑as‑text to values (Data → Text to Columns → Finish).
Multiple variables: Put each variable in its own column; check “Labels in first row.”
Outliers/skewed data: Report median (IQR) in addition to mean ± stdev.
Population metrics: ToolPak outputs sample stdev/variance; for population use STDEV.P, VAR.P.
Confidence Level value: It’s the margin of error, not the endpoints. Write CI as “mean ± margin”.