Mean = Sum of Values / Number of Values

or

x̄ = Σx / n

What it means: The arithmetic average of all values

• Add all numbers together

• Divide by how many numbers there are

• Most affected by extreme values (outliers)

If n is ODD: Median = Middle value

(n + 1) / 2 position

If n is EVEN: Median = Average of two middle values

(n/2 value + n/2+1 value) / 2

What it means: The middle number when data is ordered

• MUST arrange data in order first (ascending or descending)

• Not affected by extreme values

• Best measure when data has outliers

Mode = Most frequently occurring value

What it means: The value that appears most often

• Can have one mode (unimodal)

• Can have two modes (bimodal)

• Can have more than two modes (multimodal)

• Can have NO mode (all values appear once)

Range = Maximum - Minimum

What it means: The difference between largest and smallest values

• Shows how spread out the data is

Data: 5, 8, 3, 9, 5, 12, 5, 7

To find missing value when mean is given:

• Multiply mean by number of values to get sum

• Subtract known values from total sum

• Remainder is the missing value

Problem: The mean of 6, 8, x, 10, 12 is 9. Find x.

Adding a value GREATER than the mean:

• Mean INCREASES

• Median may or may not change

• Range may increase

Adding a value LESS than the mean:

• Mean DECREASES

• Median may or may not change

• Range may increase

Adding the SAME value to all data points:

• Mean increases by that value

• Median increases by that value

• Mode increases by that value

• Range STAYS THE SAME

Multiplying all values by the same number:

All measures multiply by that same number

MAD measures the AVERAGE DISTANCE

of each data point from the mean

Shows how spread out the data is around the mean

MAD = Σ|x - x̄| / n

Where:

x = each data value

x̄ = mean

n = number of values

| | = absolute value (always positive)

Step 1: Calculate the mean

Step 2: Find the absolute deviation of each value from the mean

Step 3: Add all absolute deviations

Step 4: Divide by the number of values

Data: 4, 6, 8, 10, 12

Quartiles divide data into FOUR equal parts

• Q₁ (Lower Quartile): 25th percentile - median of lower half

• Q₂ (Median): 50th percentile - middle value

• Q₃ (Upper Quartile): 75th percentile - median of upper half

IQR = Q₃ - Q₁

The range of the MIDDLE 50% of data

What IQR tells us:

• Shows spread of middle data

• Not affected by extreme values

• Larger IQR = more spread out data

Data (ordered): 2, 4, 6, 8, 10, 12, 14, 16, 18

An outlier is a value that is

SIGNIFICANTLY DIFFERENT from other values

Much higher or much lower than the rest of the data

Lower Fence = Q₁ - 1.5 × IQR

Upper Fence = Q₃ + 1.5 × IQR

A value is an OUTLIER if:

• It is LESS than the lower fence, OR

• It is GREATER than the upper fence

Data: Q₁ = 5, Q₃ = 15, IQR = 10. Is 32 an outlier?

POPULATION:

The ENTIRE group you want to study

Example: All students in a school

SAMPLE:

A SMALLER group selected from the population

Example: 50 students randomly chosen from the school

RANDOM SAMPLE:

Every member has EQUAL chance of being selected

REPRESENTATIVE SAMPLE:

Accurately reflects characteristics of the population

BIASED SAMPLE:

Does NOT represent the population fairly

Favors certain members over others

Use to compare typical values:

• Mean

• Median

• Mode

Use to compare variability:

• Range

• IQR (Interquartile Range)

• MAD (Mean Absolute Deviation)

When comparing:

• Higher center = higher typical values

• Higher spread = more variability/consistency

• Lower spread = more consistent data

✓ Mean: Sum divided by count, affected by outliers

✓ Median: Middle value (MUST order first), not affected by outliers

✓ Mode: Most frequent, can have multiple or none

✓ Range: Maximum minus minimum

✓ MAD: Average distance from mean (use absolute values)

✓ IQR: Q₃ - Q₁, measures middle 50%

✓ Outlier test: Use 1.5 × IQR rule with fences

✓ Population: Entire group being studied

✓ Sample: Subset of population

✓ Random sample: Everyone has equal chance

✓ Biased sample: Doesn't represent population fairly

Mean:

"Add them all, divide by the count - that's the mean, no doubt!"

Median:

"Order them first from small to great - the middle value is your mate!"

Mode:

"Mode is the most - it appears the most!"

Range:

"Max minus min - that's where range begins!"

MAD:

"Find the mean, then deviations you see - average them all for MAD, you agree!"

IQR:

"Q3 minus Q1 is IQR done - the middle half, all in one!"

Outliers:

"1.5 times IQR, add or subtract with care - beyond the fences, outliers are there!"