A statistical question is one that

expects VARIABILITY in the answers

• Can be answered by collecting DATA

• Answers will VARY from person to person

• More than ONE possible answer

Mean = Sum of all values ÷ Number of values

or

x̄ = (∑x) ÷ n

Step 1: Put data in ORDER (smallest to largest)

Step 2: Find the MIDDLE value

• If ODD number of values: middle one is median

• If EVEN number: average the two middle values

Mode = Value that appears MOST OFTEN

• Can have NO mode (all different)

• Can have MULTIPLE modes (tie)

Range = Maximum − Minimum

Largest value − Smallest value

Data: 5, 8, 3, 8, 10, 6

MAD measures how SPREAD OUT the data is

from the mean

Average distance of each value from the mean

MAD = (Sum of |each value − mean|) ÷ n

Step 1: Calculate the MEAN

Step 2: Find the ABSOLUTE DIFFERENCE between each value and mean

Step 3: ADD all the absolute differences

Step 4: DIVIDE by the number of values

Data: 2, 4, 6, 8, 10

Quartiles divide data into FOUR equal parts

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

Q2 (Second Quartile): 50th percentile - the median

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

IQR = Q3 − Q1

IQR measures the middle 50% of data

Step 1: Put data in ORDER

Step 2: Find the MEDIAN (Q2)

Step 3: Find Q1 (median of lower half)

Step 4: Find Q3 (median of upper half)

Step 5: Calculate IQR = Q3 − Q1

Data: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

An outlier is a value that is MUCH HIGHER

or MUCH LOWER than most other values

• Stands out from the rest of the data

• Significantly different from other observations

Lower Boundary: Q1 − 1.5 × IQR

Upper Boundary: Q3 + 1.5 × IQR

Any value BELOW lower boundary or ABOVE upper boundary is an outlier

• Mean: Usually changes SIGNIFICANTLY (very sensitive to outliers)

• Median: Usually changes LITTLE (resistant to outliers)

• Mode: May not change at all

• Range: Often decreases significantly

Data: 5, 6, 7, 8, 8, 9, 50

Mean: Average - use when data is symmetric, no outliers

Median: Middle value - use when data has outliers or is skewed

Mode: Most frequent - use for categorical data

Range: Max − Min (affected by outliers)

IQR: Q3 − Q1 (resistant to outliers)

MAD: Average distance from mean

Symmetric: Both sides of center are mirror images

Skewed Right: Tail extends to the right (higher values)

Skewed Left: Tail extends to the left (lower values)

When describing a distribution, mention:

• Center: What is the typical value? (mean or median)

• Spread: How spread out is the data? (range, IQR, MAD)

• Shape: Is it symmetric or skewed?

• Outliers: Are there any unusual values?

Every member of the population has an

EQUAL CHANCE of being selected

Example: Drawing names from a hat

Sample that ACCURATELY REFLECTS

the characteristics of the whole population

Example: Same proportion of males/females as population

Sample that does NOT represent the population

Results are UNFAIR or INACCURATE

Example: Only surveying people at a gym about exercise

✓ Statistical questions expect VARIABILITY in answers

✓ Mean is sensitive to outliers, median is resistant

✓ Always put data in ORDER to find median and quartiles

✓ MAD uses ABSOLUTE VALUES (no negatives)

✓ IQR = Q3 − Q1 (middle 50% of data)

✓ Outliers: Use 1.5 × IQR rule to identify

✓ Use median and IQR when outliers are present

✓ Describe distributions with center, spread, and shape

✓ Random samples give every member equal chance

✓ Biased samples lead to inaccurate conclusions

Mean, Median, Mode:

"Mean is average, Median is middle, Mode is most!"

Range:

"Biggest MINUS smallest - that's the range, oh so nicest!"

MAD:

"Mean Absolute Deviation - find each distance, no hesitation!"

IQR:

"Q3 minus Q1 - middle 50% when you're done!"

Quartiles:

"Quartiles divide in FOUR parts - Q1, Q2, Q3 from the start!"

Statistical Questions:

"If answers VARY, it's statistical - if just ONE answer, that's categorical!"