Grade 9 Exam > Grade 9 Notes > Statistics & Probability > Cheatsheet: Displaying and Comparing Quantitative Data
Cheatsheet: Displaying and Comparing Quantitative Data
1. Dotplots
1.1 Definition and Structure
| Component | Description |
|---|---|
| Dotplot | Graph showing each data value as a dot above a number line |
| Structure | Horizontal axis represents data values; dots stacked vertically for repeated values |
1.2 When to Use
- Small to moderate-sized datasets (n <>
- Showing individual data points while revealing distribution shape
- Comparing two or more small groups side-by-side
1.3 Advantages and Limitations
| Advantages | Limitations |
|---|---|
| Preserves individual values; easy to identify clusters, gaps, and outliers | Impractical for large datasets; becomes cluttered with many observations |
2. Stemplots (Stem-and-Leaf Plots)
2.1 Definition and Construction
| Term | Definition |
|---|---|
| Stemplot | Display separating each value into a stem (leading digit(s)) and leaf (trailing digit) |
| Stem | Leading digit(s) written vertically on left side |
| Leaf | Trailing digit written horizontally to the right of corresponding stem |
2.2 Types of Stemplots
| Type | Description |
|---|---|
| Regular Stemplot | Each stem appears once (e.g., 2 | 3 5 7 represents 23, 25, 27) |
| Split Stemplot | Each stem split into two rows: first for leaves 0-4, second for leaves 5-9 |
| Back-to-Back Stemplot | Two distributions sharing same stem in middle; leaves extend left and right |
2.3 Key Properties
- Leaves written in ascending order for each stem
- Preserves actual data values (can reconstruct original dataset)
- Shows distribution shape, center, and spread
- Effective for datasets with 15-150 observations
- Include key or legend explaining stem and leaf units
3. Histograms
3.1 Definition and Components
| Component | Description |
|---|---|
| Histogram | Bar graph showing frequency or relative frequency of quantitative data in intervals |
| Bin (Class) | Interval of values on horizontal axis |
| Frequency | Count of observations falling within each bin |
| Relative Frequency | Proportion of observations in each bin (frequency ÷ total n) |
3.2 Construction Rules
- Bins must be equal width
- Bins are contiguous with no gaps
- Each observation falls into exactly one bin
- Bars touch each other (no spaces between bars)
- Height represents frequency or relative frequency
- Use 5-20 bins depending on dataset size
3.3 Key Features
| Feature | Description |
|---|---|
| Bin Width | Calculated as (max - min) ÷ number of bins |
| Area Interpretation | In relative frequency histogram, total area equals 1 |
| Individual Values | Not preserved; only see distribution pattern |
4. Cumulative Relative Frequency Plots
4.1 Definition
| Term | Definition |
|---|---|
| Cumulative Relative Frequency | Sum of relative frequencies up to and including current bin |
| Cumulative Plot | Graph showing cumulative relative frequency against upper bin boundaries |
4.2 Properties
- Plotted as line graph or step function
- Always increases from left to right (non-decreasing)
- Starts at 0 and ends at 1 (or 100%)
- Useful for finding percentiles and proportions below specific values
5. Distribution Shapes
5.1 Symmetry
| Shape | Description |
|---|---|
| Symmetric | Left and right sides mirror each other around center |
| Skewed Right (Positive Skew) | Tail extends toward higher values; bulk of data on left |
| Skewed Left (Negative Skew) | Tail extends toward lower values; bulk of data on right |
5.2 Modality
| Type | Description |
|---|---|
| Unimodal | One clear peak or mode |
| Bimodal | Two distinct peaks |
| Multimodal | More than two peaks |
| Uniform | No clear peak; all values roughly equal frequency |
5.3 Other Features
- Gaps: Intervals with no data between groups of observations
- Clusters: Groups of data points separated from others
- Outliers: Individual values far from the bulk of data
6. Center Measures
6.1 Mean
| Term | Description |
|---|---|
| Mean (x̄) | Sum of all values divided by number of observations: x̄ = (Σx) ÷ n |
| Properties | Sensitive to outliers and skewness; balancing point of distribution |
6.2 Median
| Term | Description |
|---|---|
| Median | Middle value when data ordered; divides data into two equal halves |
| Calculation (odd n) | Position = (n+1) ÷ 2 |
| Calculation (even n) | Average of values at positions n/2 and (n/2)+1 |
| Properties | Resistant to outliers; better for skewed distributions |
6.3 Mode
| Term | Description |
|---|---|
| Mode | Most frequently occurring value(s) |
| Properties | May have no mode, one mode, or multiple modes |
6.4 Choosing Appropriate Measure
- Use median for skewed distributions or when outliers present
- Use mean for symmetric distributions without outliers
- Mean > Median indicates right skew; Mean < median="" indicates="" left="">
7. Spread Measures
7.1 Range
| Term | Definition |
|---|---|
| Range | Difference between maximum and minimum values: Range = Max - Min |
| Properties | Simple but sensitive to outliers; only uses two values |
7.2 Interquartile Range (IQR)
| Term | Definition |
|---|---|
| Q1 (First Quartile) | Median of lower half of data; 25th percentile |
| Q3 (Third Quartile) | Median of upper half of data; 75th percentile |
| IQR | Spread of middle 50% of data: IQR = Q3 - Q1 |
| Properties | Resistant to outliers; used with median |
7.3 Variance and Standard Deviation
| Measure | Formula and Description |
|---|---|
| Variance (s²) | s² = Σ(x - x̄)² ÷ (n-1); average squared deviation from mean |
| Standard Deviation (s) | s = √[Σ(x - x̄)² ÷ (n-1)]; square root of variance |
| Properties | Both sensitive to outliers; use with mean; same units as original data (for s) |
7.4 Outlier Identification
| Method | Definition |
|---|---|
| 1.5 × IQR Rule | Outlier if below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR) |
| Lower Fence | Q1 - 1.5(IQR) |
| Upper Fence | Q3 + 1.5(IQR) |
8. Boxplots (Box-and-Whisker Plots)
8.1 Components
| Component | Description |
|---|---|
| Box | Rectangle from Q1 to Q3; contains middle 50% of data |
| Line in Box | Marks the median (Q2) |
| Whiskers | Lines extending from box to smallest and largest non-outlier values |
| Outliers | Individual points plotted beyond whiskers |
8.2 Five-Number Summary
- Minimum (excluding outliers in modified boxplot)
- Q1 (First Quartile)
- Median (Q2)
- Q3 (Third Quartile)
- Maximum (excluding outliers in modified boxplot)
8.3 Types of Boxplots
| Type | Description |
|---|---|
| Standard Boxplot | Whiskers extend to minimum and maximum values |
| Modified Boxplot | Whiskers extend to furthest non-outlier; outliers plotted separately |
| Side-by-Side Boxplots | Multiple boxplots on same scale for comparing groups |
8.4 Interpreting Boxplots
- Longer box indicates greater spread in middle 50%
- Median closer to Q1: right skewed; closer to Q3: left skewed
- Longer whisker on one side suggests skewness in that direction
- Effective for comparing distributions across multiple groups
9. Comparing Distributions
9.1 Framework for Comparison
| Aspect | What to Compare |
|---|---|
| Shape | Symmetry vs. skewness, modality, presence of gaps or clusters |
| Center | Compare means or medians; which group has higher typical values |
| Spread | Compare IQR or standard deviation; which group is more variable |
| Outliers | Presence, number, and direction of unusual values |
9.2 Comparison Displays
| Display Type | Best Use |
|---|---|
| Side-by-Side Boxplots | Comparing center, spread, and outliers across multiple groups |
| Back-to-Back Stemplots | Comparing two groups while preserving individual values |
| Multiple Histograms | Comparing shape and distribution patterns; use same scale and bin width |
| Comparative Dotplots | Small datasets where individual values matter |
9.3 Context Considerations
- Always use same scale on axis for valid visual comparison
- Consider practical significance, not just numerical differences
- Relate findings to context of data (units, subject matter)
10. Percentiles and Standardized Scores
10.1 Percentiles
| Term | Definition |
|---|---|
| pth Percentile | Value below which p% of data falls |
| Quartiles | Q1 = 25th percentile; Q2 = 50th percentile (median); Q3 = 75th percentile |
| Interpretation | Describes relative position within distribution |
10.2 Z-scores (Standardized Scores)
| Component | Description |
|---|---|
| Z-score Formula | z = (x - x̄) ÷ s |
| Interpretation | Number of standard deviations a value is from the mean |
| Positive z-score | Value above the mean |
| Negative z-score | Value below the mean |
| z = 0 | Value equals the mean |
10.3 Uses of Z-scores
- Comparing values from different distributions or units
- Identifying unusual values (|z| > 2 or |z| > 3 considered unusual)
- Standardizing data to have mean = 0 and standard deviation = 1
11. Effects of Transformations on Data
11.1 Adding/Subtracting a Constant
| Transformation | Effect |
|---|---|
| y = x + c | Adds c to mean and median; no change to spread (s, IQR, range) |
| Shape | Unchanged; distribution shifts horizontally |
11.2 Multiplying/Dividing by a Constant
| Transformation | Effect |
|---|---|
| y = cx | Multiplies mean, median, s, IQR, and range by |c| |
| Shape | Unchanged; distribution stretches or compresses horizontally |
11.3 Combined Linear Transformations
- For y = a + bx: mean(y) = a + b×mean(x)
- Standard deviation(y) = |b| × standard deviation(x)
- Shape remains unchanged
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