1. Mean, Median, Mode, and Range

These are measures of central tendency and spread that help us understand and describe data.

Mean (Average):

\( \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{\sum x}{n} \)

Add all values and divide by how many there are

Median (Middle Value):

• Step 1: Arrange data in order from least to greatest
• Step 2: Find the middle value:
  • Odd number of values: Middle value is the median
  • Even number of values: Average the two middle values

Mode (Most Frequent):

The value that appears most often in the data set

• Can have one mode (unimodal)
• Can have multiple modes (bimodal, multimodal)
• Can have no mode if all values appear equally

Range (Spread):

\( \text{Range} = \text{Maximum value} - \text{Minimum value} \)

Example:

Data set: 12, 15, 18, 15, 22, 30, 15

Mean: \( \frac{12 + 15 + 18 + 15 + 22 + 30 + 15}{7} = \frac{127}{7} = 18.14 \)

Median: Order: 12, 15, 15, 15, 18, 22, 30 → Middle value = 15

Mode: 15 (appears 3 times)

Range: 30 - 12 = 18

2. Interpret Charts and Graphs

Common Chart Types:

Frequency Tables: Shows how often each value appears

• Multiply each value by its frequency, then sum
• Divide by total frequency to get mean

Dot Plots: Each dot represents one data point

• Count dots at each value
• Most dots = mode

Bar Graphs: Height of bars shows frequency

• Read frequency from y-axis
• Tallest bar = mode

Example: Frequency Table

Mean: \( \frac{(5×3) + (7×5) + (9×2)}{3+5+2} = \frac{15+35+18}{10} = \frac{68}{10} = 6.8 \)

Mode: 7 (highest frequency of 5)

3. Find the Missing Number

Strategy for Finding Missing Value:

When given the mean:

1. Multiply mean by number of values to get total sum
2. Subtract known values from total sum
3. Result is the missing value

When given the median:

1. Arrange known values in order
2. Determine where missing value must be placed
3. Use median position to find missing value

Example 1: Finding Missing Value (Mean)

Problem: The mean of 15, 20, x, 25 is 22. Find x.

Step 1: Total sum = Mean × Number of values = 22 × 4 = 88

Step 2: Sum of known values = 15 + 20 + 25 = 60

Step 3: x = 88 - 60 = 28

Answer: x = 28

Example 2: Finding Missing Value (Median)

Problem: The median of 8, 12, x, 20, 24 is 15. Find x.

Step 1: For 5 values, median is the 3rd value

Step 2: In order: 8, 12, ?, ?, ?

Step 3: The middle value must be 15, so x = 15

Answer: x = 15

4. Changes in Mean, Median, Mode, and Range

Effects of Adding/Removing a Value:

Example:

Original data: 10, 12, 15, 15, 18

Mean = 14, Median = 15, Mode = 15, Range = 8

Add 30 to the data: 10, 12, 15, 15, 18, 30

Mean = 16.67 (increased), Median = 15 (same), Mode = 15 (same), Range = 20 (increased)

Key Observations:

• Mean is most affected by outliers
• Median is resistant to outliers
• Mode only changes if frequencies change
• Range changes when min or max changes

5. Mean Absolute Deviation (MAD)

Definition: The average distance of each data point from the mean. It measures how spread out the data is.

Formula:

\( \text{MAD} = \frac{\sum |x_i - \bar{x}|}{n} \)

where \( x_i \) = each data value, \( \bar{x} \) = mean, \( n \) = number of values

Steps to Calculate MAD:

1. Find the mean of the data set
2. Find the absolute deviation of each value from the mean: \( |x_i - \bar{x}| \)
3. Add all the absolute deviations
4. Divide by the number of values

Example:

Data: 4, 8, 6, 10, 12

Step 1: Mean = \( \frac{4+8+6+10+12}{5} = \frac{40}{5} = 8 \)

Step 2: Find absolute deviations:

Step 3: Sum of absolute deviations = 4 + 0 + 2 + 2 + 4 = 12

Step 4: MAD = \( \frac{12}{5} = 2.4 \)

Answer: MAD = 2.4 (on average, values are 2.4 units from the mean)

6. Quartiles and Interquartile Range (IQR)

Quartiles:

Quartiles divide ordered data into four equal parts.

• Q₁ (First Quartile): Median of lower half (25th percentile)
• Q₂ (Second Quartile): Median of entire data (50th percentile)
• Q₃ (Third Quartile): Median of upper half (75th percentile)

Interquartile Range (IQR):

\( \text{IQR} = Q_3 - Q_1 \)

IQR measures the spread of the middle 50% of the data

Steps to Find Quartiles:

1. Arrange data in order from least to greatest
2. Find Q₂ (median of all data)
3. Find Q₁ (median of lower half, excluding Q₂)
4. Find Q₃ (median of upper half, excluding Q₂)
5. Calculate IQR = Q₃ - Q₁

Example:

Data: 3, 7, 8, 12, 13, 14, 18, 21, 22

Step 1: Already in order (9 values)

Step 2: Q₂ (median) = 13 (5th value)

Step 3: Lower half: 3, 7, 8, 12 → Q₁ = \( \frac{7+8}{2} = 7.5 \)

Step 4: Upper half: 14, 18, 21, 22 → Q₃ = \( \frac{18+21}{2} = 19.5 \)

Step 5: IQR = 19.5 - 7.5 = 12

Five-Number Summary: Min = 3, Q₁ = 7.5, Q₂ = 13, Q₃ = 19.5, Max = 22

7. Box Plots (Box-and-Whisker Plots)

Definition: A visual display of the five-number summary showing the distribution and spread of data.

Five-Number Summary:

1. Minimum: Smallest value (left whisker)
2. Q₁: First quartile (left edge of box)
3. Median (Q₂): Middle value (line inside box)
4. Q₃: Third quartile (right edge of box)
5. Maximum: Largest value (right whisker)

Parts of a Box Plot:

• Box: Contains middle 50% of data (from Q₁ to Q₃)
• Whiskers: Lines extending to minimum and maximum
• Median line: Vertical line inside the box

Reading a Box Plot:

• Width of box = IQR: Shows spread of middle 50%
• Length of whiskers: Shows overall range
• Median position: Shows if data is skewed
• Outliers: Marked as individual points beyond whiskers

Interpreting Box Plots:

8. Identify an Outlier

Definition: An outlier is a data value that is much higher or much lower than most of the other values in a data set.

Methods to Identify Outliers:

Method 1: Visual Inspection

Look for values that are far from the rest of the data

Method 2: 1.5 × IQR Rule

Lower boundary: \( Q_1 - 1.5 \times \text{IQR} \)

Upper boundary: \( Q_3 + 1.5 \times \text{IQR} \)

Any value below lower boundary or above upper boundary is an outlier

Example:

Data: 12, 15, 18, 20, 22, 25, 28, 75

Find quartiles:

Q₁ = 16.5, Q₂ = 21, Q₃ = 26.5

Calculate IQR: IQR = 26.5 - 16.5 = 10

Calculate boundaries:

Lower: 16.5 - 1.5(10) = 16.5 - 15 = 1.5

Upper: 26.5 + 1.5(10) = 26.5 + 15 = 41.5

75 > 41.5, so 75 is an outlier

9. Effect of Removing an Outlier

Impact on Statistics:

Example:

With outlier: 5, 8, 10, 12, 15, 50

Mean = 16.67, Median = 11, Mode = none, Range = 45

Without outlier: 5, 8, 10, 12, 15

Mean = 10, Median = 10, Mode = none, Range = 10

Observations:

• Mean decreased by 6.67 (40% change)
• Median decreased by only 1 (9% change)
• Range decreased by 35 (78% change)

Conclusion: Median and IQR are better measures when outliers are present.

Quick Reference: One-Variable Statistics

Key Formulas:

Mean: \( \bar{x} = \frac{\sum x}{n} \)

Range: \( \text{Max} - \text{Min} \)

MAD: \( \frac{\sum |x_i - \bar{x}|}{n} \)

IQR: \( Q_3 - Q_1 \)

Outlier boundaries: \( Q_1 - 1.5 \times \text{IQR} \) and \( Q_3 + 1.5 \times \text{IQR} \)

Five-Number Summary:

Minimum, Q₁, Median (Q₂), Q₃, Maximum

Resistant vs Sensitive Measures:

• Resistant (not affected by outliers): Median, IQR
• Sensitive (affected by outliers): Mean, Range, MAD

💡 Key Tips for One-Variable Statistics

• ✓ Mean = average (add all, divide by count)
• ✓ Median = middle value (order data first!)
• ✓ Mode = most frequent value
• ✓ Range = max - min (spread of all data)
• ✓ Always order data before finding median and quartiles
• ✓ MAD shows average distance from mean
• ✓ IQR shows spread of middle 50% of data
• ✓ Box plot shows five-number summary visually
• ✓ Outlier = value beyond 1.5 × IQR from quartiles
• ✓ Mean is sensitive to outliers; median is not
• ✓ Use median for skewed data or data with outliers
• ✓ Removing outliers: mean and range change most