Study guide
14.3 Descriptive Statistics, Data Displays, and Probability Theory
California Subject Examinations for Teachers CSET
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12 Chapter 12: Number Sense, Operations, and Fractions
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14 Chapter 14: Coordinate Geometry, Measurement, Statistics, and Mathematical Practices
14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations 14.2 Measurement Systems, Dimensional Analysis, and Precision 14.3 Descriptive Statistics, Data Displays, and Probability Theory 14.4 Mathematical Practices: Reasoning, Modeling, Estimation, and Justification
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18.1 Elements of Art, Principles of Design, and Visual Analysis 18.2 Art Media, Techniques, World Art History, and Art Criticism 18.3 California Arts Standards and Cross-Curricular Arts Integration 18.4 CSET Constructed-Response Strategy, Scoring Scale, and Domain-Specific Exemplars
14.3 Descriptive Statistics, Data Displays, and Probability Theory
Key Takeaways Explain This Topic AI
Descriptive statistics summarize data distributions through measures of central tendency (Mean x ˉ \bar{x} x ˉ , Median, Mode, Weighted Mean) and dispersion (Range, Quartiles, Interquartile Range IQR = Q 3 − Q 1 \text{IQR} = Q_3 - Q_1 IQR = Q 3 − Q 1 , Variance σ 2 \sigma^2 σ 2 , and Standard Deviation σ \sigma σ ).
Data skewness alters central tendency relationships: symmetric distributions yield Mean ≈ Median ≈ Mode \text{Mean} \approx \text{Median} \approx \text{Mode} Mean ≈ Median ≈ Mode , right-skewed data pull the mean above the median ( Mean > Median \text{Mean} > \text{Median} Mean > Median ), and left-skewed data pull the mean below the median ( Mean < Median \text{Mean} < \text{Median} Mean < Median ).
Tukey's Outlier Rule identifies data points beyond the inner fences [ Q 1 − 1.5 × IQR , Q 3 + 1.5 × IQR ] [Q_1 - 1.5 \times \text{IQR}, \; Q_3 + 1.5 \times \text{IQR}] [ Q 1 − 1.5 × IQR , Q 3 + 1.5 × IQR ] ; for normal distributions, the Empirical Rule establishes that 68.3 % 68.3\% 68.3% , 95.5 % 95.5\% 95.5% , and 99.7 % 99.7\% 99.7% of values lie within 1 1 1 , 2 2 2 , and 3 3 3 standard deviations of the mean.
Data displays serve specific analytical purposes: bar/pie charts represent categorical data, while dot plots, stem-and-leaf plots, histograms, and box-and-whisker plots represent quantitative distributions; scatter plots display bivariate correlation (which never proves causation).
Probability principles govern random events: Complement Rule ( P ( E ′ ) = 1 − P ( E ) P(E')=1-P(E) P ( E ′ ) = 1 − P ( E ) ), Addition Rule ( P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) P(A \cup B) = P(A)+P(B)-P(A \cap B) P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) ), and Multiplication Rule ( P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A ) P(A \cap B) = P(A) \cdot P(B|A) P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A ) ); combinatorics distinguishes permutations ( n P r nPr n P r , order matters) from combinations ( n C r nCr n C r , order does not matter).
Last updated: August 2026
14.3 Descriptive Statistics, Data Displays, and Probability Theory
CSET Focus: Statistics and probability on CSET Multiple Subjects Subtest II test your ability to calculate and interpret summary statistics, evaluate how skewness affects mean versus median, apply the 1.5 × IQR 1.5 \times \text{IQR} 1.5 × IQR outlier rule, interpret graphical data displays (including boxplots and histograms), solve single and compound probability problems with and without replacement, and apply counting principles (permutations vs. combinations) to classroom scenarios.
1. Descriptive Statistics: Central Tendency, Spread, and Distribution Shape
Descriptive statistics synthesize raw numerical datasets into meaningful quantitative summaries characterizing central location, spread, and distributional symmetry. DESCRIPTIVE STATISTICS
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- MEASURES OF CENTRAL TENDENCY MEASURES OF DISPERSION (SPREAD)
- Mean (x̄): Arithmetic average • Range: Max - Min (Sensitive to outliers)
- Median (Q₂): Middle value (Resistant) • IQR: Q₃ - Q₁ (Middle 50% spread, Resistant)
- Mode: Most frequent value (Categorical) • Variance (s²): Mean squared deviation
- Weighted Mean: Value weighting by frequency • Standard Deviation (s): Root mean squared spread
- Measures of Central Tendency
Arithmetic Mean ( x ˉ \bar{x} x ˉ or μ \mu μ ): The sum of all observations divided by the total number of observations: x ˉ = ∑ i = 1 n x i n \bar{x} = \frac{\sum_{i=1}^n x_i}{n} x ˉ = n ∑ i = 1 n x i
Property: Highly sensitive to extreme values and outliers.
Weighted Mean ( x ˉ w \bar{x}_w x ˉ w ): Computes an average where each value is weighted by its relative importance or frequency w i w_i w i : x ˉ w = ∑ i = 1 k w i x i ∑ i = 1 k w i \bar{x}_w = \frac{\sum_{i=1}^k w_i x_i}{\sum_{i=1}^k w_i} x ˉ w = ∑ i = 1 k w i ∑ i = 1 k w i x i
Example: Course grade calculation with 3 3 3 tests at 20 % 20\% 20% each ( 60 % 60\% 60% total) and a final exam at 40 % 40\% 40% .
Median ( Q 2 Q_2 Q 2 / x ~ \tilde{x} x ~ ): The physical middle value in an ordered (sorted) dataset:
If n n n is odd, the median is the unique value at position n + 1 2 \frac{n+1}{2} 2 n + 1 .
If n n n is even, the median is the arithmetic mean of the two middle values at positions n 2 \frac{n}{2} 2 n and n 2 + 1 \frac{n}{2} + 1 2 n + 1 .
Property: Resistant (robust) measure of center; not influenced by extreme outliers.
Mode: The data value(s) occurring with the highest frequency. A distribution may be unimodal (one mode), bimodal (two modes), multimodal, or have no mode (if all values occur equally). The mode is the only measure of central tendency appropriate for qualitative/nominal data.
Skewness and the Mean-Median Relationship Left-Skewed (Negatively Skewed) Symmetric (Bell-Shaped) Right-Skewed (Positively Skewed)
- Long Left Tail Long Right Tail
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- Mean < Median < Mode Mean ≈ Median ≈ Mode Mode < Median < Mean
Symmetric Distribution: The data are balanced around a central peak. Mean ≈ Median ≈ Mode \text{Mean} \approx \text{Median} \approx \text{Mode} Mean ≈ Median ≈ Mode .
Right-Skewed (Positively Skewed): Tail extends toward higher values to the right. Extreme high values pull the mean upward: Mean > Median > Mode \mathbf{\text{Mean} > \text{Median} > \text{Mode}} Mean > Median > Mode . (Example: Household income, home sale prices). Use Median and IQR as the preferred summary statistics.
Left-Skewed (Negatively Skewed): Tail extends toward lower values to the left. Extreme low values pull the mean downward: Mean < Median < Mode \mathbf{\text{Mean} < \text{Median} < \text{Mode}} Mean < Median < Mode . (Example: Age at natural retirement, easy exam scores). Use Median and IQR .
Measures of Dispersion and the Five-Number Summary
Range: Difference between the maximum and minimum observations: Range = x max − x min \text{Range} = x_{\max} - x_{\min} Range = x m a x − x m i n .
Quartiles and Interquartile Range (IQR):
First Quartile ( Q 1 Q_1 Q 1 ): 25th percentile (median of the lower half of data).
Second Quartile ( Q 2 Q_2 Q 2 ): 50th percentile (median of the entire dataset).
Third Quartile ( Q 3 Q_3 Q 3 ): 75th percentile (median of the upper half of data).
Interquartile Range (IQR): The spread of the middle 50 % 50\% 50% of data: IQR = Q 3 − Q 1 \mathbf{\text{IQR} = Q_3 - Q_1} IQR = Q 3 − Q 1 . (Resistant to outliers).
The Five-Number Summary: Comprises [ Minimum , Q 1 , Q 2 , Q 3 , Maximum ] [\text{Minimum}, \; Q_1, \; Q_2, \; Q_3, \; \text{Maximum}] [ Minimum , Q 1 , Q 2 , Q 3 , Maximum ] .
Tukey's Outlier Detection Rule ( 1.5 × IQR 1.5 \times \text{IQR} 1.5 × IQR Criterion)
A data observation x x x is formally classified as an outlier if it lies outside the inner fences: Lower Fence = Q 1 − 1.5 × IQR Upper Fence = Q 3 + 1.5 × IQR \text{Lower Fence} = Q_1 - 1.5 \times \text{IQR} \qquad \text{Upper Fence} = Q_3 + 1.5 \times \text{IQR} Lower Fence = Q 1 − 1.5 × IQR Upper Fence = Q 3 + 1.5 × IQR
Outlier Criterion: x < Lower Fence x < \text{Lower Fence} x < Lower Fence OR x > Upper Fence x > \text{Upper Fence} x > Upper Fence .
Standard Deviation & The Empirical Rule (68-95-99.7%)
Sample Variance ( s 2 s^2 s 2 ) and Standard Deviation ( s s s ): s 2 = ∑ i = 1 n ( x i − x ˉ ) 2 n − 1 s = ∑ i = 1 n ( x i − x ˉ ) 2 n − 1 s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1} \qquad s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}} s 2 = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2 s = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2
The Empirical Rule (68-95-99.7% Rule for Normal Distributions):
Approximately 68.27 % 68.27\% 68.27% of data fall within μ ± 1 σ \mu \pm 1\sigma μ ± 1 σ .
Approximately 95.45 % 95.45\% 95.45% of data fall within μ ± 2 σ \mu \pm 2\sigma μ ± 2 σ .
Approximately 99.73 % 99.73\% 99.73% of data fall within μ ± 3 σ \mu \pm 3\sigma μ ± 3 σ .
Standard Score ( z z z -score): Measures the signed distance from the mean in standard deviation units: z = x − μ σ z = \frac{x - \mu}{\sigma} z = σ x − μ .
- 2. Graphical Data Displays and Bivariate Analysis DATA DISPLAYS
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- CATEGORICAL DISPLAYS QUANTITATIVE DISPLAYS
- Bar Charts (Spaces between bars) • Dot Plots / Line Plots (Discrete points on line)
- Pie Charts / Circle Graphs (Angle = % × 360°) • Stem-and-Leaf Plots (Preserves exact raw values)
- Histograms (Continuous contiguous bins, no gaps)
- Box-and-Whisker Plots (5-number summary & outliers)
- Scatter Plots (Bivariate correlation & trend lines)
- Graphical Data Displays Matrix
Display Type Data Type Primary Visual Features Analytical Strengths Common Misinterpretations
Bar Chart Categorical / Discrete Separated vertical/horizontal bars Compares distinct category counts Confusing with histograms; omitting zero baseline
Pie Chart Categorical Proportional circular sectors; θ = f N × 360 ∘ \theta = \frac{f}{N} \times 360^\circ θ = N f × 36 0 ∘ Shows part-to-whole percentage distribution Comparing slices without percentage labels; > 100 % >100\% > 100% totals
Dot Plot Quantitative (Discrete) Stacked dots along a linear axis Shows exact values, clusters, gaps, and modes Overwhelming with large sample sizes ( n > 100 n > 100 n > 100 )
Stem-and-Leaf Quantitative Leading digit stem with ordered trailing leaves Preserves exact individual data values while showing shape Forgetting key/legend (e.g., 4 ∣ 2 = 42 4 \mid 2 = 42 4 ∣ 2 = 42 ); misaligning columns
Histogram Quantitative (Continuous) Contiguous adjacent bars over equal bin intervals Visualizes distribution shape, skewness, and spread Assuming bar width represents category rather than interval range
Box Plot Quantitative Five-number summary box ( Q 1 Q_1 Q 1 to Q 3 Q_3 Q 3 ) with whiskers Rapidly identifies median, IQR, skewness, and outliers Misinterpreting whisker length as frequency rather than spread
Scatter Plot Bivariate Quantitative Paired ( x , y ) (x, y) ( x , y ) coordinate points Reveals correlation, linearity, clusters, and regression Confusing correlation with causation ; extrapolation error
Bivariate Data: Scatter Plots and Correlation vs. Causation
Scatter Plots: Display paired quantitative data ( x i , y i ) (x_i, y_i) ( x i , y i ) to investigate associative relationships between an independent explanatory variable ( x x x ) and a dependent response variable ( y y y ).
Linear Correlation Coefficient ( r r r ): Measures the strength and direction of a linear relationship ( − 1 ≤ r ≤ 1 -1 \le r \le 1 − 1 ≤ r ≤ 1 ):
r = + 1 r = +1 r = + 1 : Perfect positive linear correlation.
r = − 1 r = -1 r = − 1 : Perfect negative linear correlation.
r = 0 r = 0 r = 0 : No linear correlation.
Trend Line / Line of Best Fit ( y = m x + b y = mx + b y = m x + b ): Used for interpolation (estimating within the observed data domain) and cautious extrapolation (predicting outside the observed domain).
Correlation vs. Causation: A strong statistical correlation between two variables does NOT imply a cause-and-effect relationship. Apparent correlations are frequently driven by confounding or lurking variables (e.g., shoe size and reading ability correlate strongly in elementary school children due to the lurking variable of age).
3. Probability Theory and Combinatorics
Probability quantifies the likelihood of an event occurring within a well-defined sample space ( S S S ).
Foundations of Probability
Theoretical Probability: Based on mathematical analysis of equally likely outcomes in sample space S S S : P ( E ) = n ( E ) n ( S ) = Number of favorable outcomes Total number of possible outcomes P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} P ( E ) = n ( S ) n ( E ) = Total number of possible outcomes Number of favorable outcomes
Experimental (Empirical) Probability: Based on observed relative frequency from empirical trials: P exp ( E ) = Frequency of event occurrence Total number of experimental trials P_{\text{exp}}(E) = \frac{\text{Frequency of event occurrence}}{\text{Total number of experimental trials}} P exp ( E ) = Total number of experimental trials Frequency of event occurrence
The Law of Large Numbers: As the number of independent repetitions ( N N N ) increases, the empirical experimental probability converges toward the theoretical probability: lim N → ∞ P exp ( E ) = P ( E ) \lim_{N \to \infty} P_{\text{exp}}(E) = P(E) lim N → ∞ P exp ( E ) = P ( E ) .
Fundamental Probability Axioms & Operational Rules
Probability Bounds: 0 ≤ P ( E ) ≤ 1 0 \le P(E) \le 1 0 ≤ P ( E ) ≤ 1 . Impossible event: P ( ∅ ) = 0 P(\emptyset) = 0 P ( ∅ ) = 0 ; Certain event: P ( S ) = 1 P(S) = 1 P ( S ) = 1 .
Complement Rule: The probability that event E E E does not occur ( E ′ E' E ′ or E c E^c E c ): P ( E ′ ) = 1 − P ( E ) P(E') = 1 - P(E) P ( E ′ ) = 1 − P ( E )
"At Least One" Strategy: To find the probability of at least one success across multiple trials: P ( At least one ) = 1 − P ( None ) P(\text{At least one}) = 1 - P(\text{None}) P ( At least one ) = 1 − P ( None )
Addition Rule (Union of Events):
Mutually Exclusive (Disjoint) Events ( A ∩ B = ∅ A \cap B = \emptyset A ∩ B = ∅ ): P ( A ∪ B ) = P ( A ) + P ( B ) P(A \cup B) = P(A) + P(B) P ( A ∪ B ) = P ( A ) + P ( B )
Non-Mutually Exclusive (Inclusive) Events ( A ∩ B ≠ ∅ A \cap B \ne \emptyset A ∩ B = ∅ ): P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) P(A \cup B) = P(A) + P(B) - P(A \cap B) P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B )
Conditional Probability & Multiplication Rule (Intersection of Events):
Conditional Probability ( P ( B ∣ A ) P(B|A) P ( B ∣ A ) ): The probability of B B B occurring given that A A A has already occurred: P ( B ∣ A ) = P ( A ∩ B ) P ( A ) ( P ( A ) > 0 ) P(B|A) = \frac{P(A \cap B)}{P(A)} \quad (P(A) > 0) P ( B ∣ A ) = P ( A ) P ( A ∩ B ) ( P ( A ) > 0 )
General Multiplication Rule: P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A ) P(A \cap B) = P(A) \cdot P(B|A) P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A )
Independent Events: Two events are independent if P ( B ∣ A ) = P ( B ) P(B|A) = P(B) P ( B ∣ A ) = P ( B ) . Then: P ( A ∩ B ) = P ( A ) ⋅ P ( B ) P(A \cap B) = P(A) \cdot P(B) P ( A ∩ B ) = P ( A ) ⋅ P ( B )
Dependent Events (Sampling Without Replacement): Probabilities change on each successive draw because the sample space decreases.
- Combinatorics: Counting Principles, Permutations, and Combinations COUNTING TECHNIQUES
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- ORDER MATTERS? YES ORDER MATTERS? NO
- PERMUTATIONS (nPr) COMBINATIONS (nCr)
- Arranging officers (Pres, VP, Sec) • Choosing committees, teams, hands
- Seating in a line, race finish order • Selecting toppings, fruit salads
- nPr = n! / (n - r)! • nCr = n! / [r!(n - r)!]
Fundamental Counting Principle: If a multi-step task consists of k k k independent successive stages with n 1 , n 2 , … , n k n_1, n_2, \dots, n_k n 1 , n 2 , … , n k choices, total outcomes = n 1 × n 2 × ⋯ × n k = n_1 \times n_2 \times \dots \times n_k = n 1 × n 2 × ⋯ × n k .
Factorial Notation ( n ! n! n ! ): For n ∈ Z + n \in \mathbb{Z}^+ n ∈ Z + , n ! = n ( n − 1 ) ( n − 2 ) ⋯ ( 1 ) n! = n(n-1)(n-2)\cdots(1) n ! = n ( n − 1 ) ( n − 2 ) ⋯ ( 1 ) , with the formal convention 0 ! = 1 0! = 1 0 ! = 1 .
Permutations ( n P r nPr n P r — Order Matters): The number of ordered arrangements of r r r items selected from a set of n n n distinct items: n P r = n ! ( n − r ) ! _n P_r = \frac{n!}{(n - r)!} n P r = ( n − r )! n !
Example: Choosing 1st, 2nd, and 3rd place winners from 8 runners: 8 P 3 = 8 ! ( 8 − 3 ) ! = 8 ! 5 ! = 8 × 7 × 6 = 336 _8 P_3 = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 8 \times 7 \times 6 = 336 8 P 3 = ( 8 − 3 )! 8 ! = 5 ! 8 ! = 8 × 7 × 6 = 336
Combinations ( n C r nCr n C r or ( n r ) \binom{n}{r} ( r n ) — Order Does NOT Matter): The number of unordered subsets of r r r items chosen from a set of n n n distinct items: n C r = ( n r ) = n ! r ! ( n − r ) ! = n P r r ! _n C_r = \binom{n}{r} = \frac{n!}{r!(n - r)!} = \frac{_n P_r}{r!} n C r = ( r n ) = r ! ( n − r )! n ! = r ! n P r
Example: Selecting a committee of 3 teachers from a department of 8 teachers: 8 C 3 = 8 ! 3 ! ( 8 − 3 ) ! = 8 × 7 × 6 3 × 2 × 1 = 56 _8 C_3 = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 8 C 3 = 3 ! ( 8 − 3 )! 8 ! = 3 × 2 × 1 8 × 7 × 6 = 56
Probability & Combinatorics Reference Matrix
Concept / Rule Mathematical Formula Context Clues / Key Question Phrases Exemplar Problem Setup
Complement P ( E ′ ) = 1 − P ( E ) P(E') = 1 - P(E) P ( E ′ ) = 1 − P ( E ) "Probability of not", "at least one" P ( ≥ 1 Head in 4 flips ) = 1 − ( 1 / 2 ) 4 = 15 / 16 P(\ge 1\text{ Head in 4 flips}) = 1 - (1/2)^4 = 15/16 P ( ≥ 1 Head in 4 flips ) = 1 − ( 1/2 ) 4 = 15/16
Addition Rule P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) P(A \cup B) = P(A)+P(B)-P(A \cap B) P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) "Probability of A A A OR B B B " Card is King or Heart: 4 / 52 + 13 / 52 − 1 / 52 = 16 / 52 4/52 + 13/52 - 1/52 = 16/52 4/52 + 13/52 − 1/52 = 16/52
Multiplication Rule P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A ) P(A \cap B) = P(A) \cdot P(B\vert A) P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A ) "Probability of A A A AND B B B ", sequential draws 2 Aces without replacement: ( 4 / 52 ) × ( 3 / 51 ) = 1 / 221 (4/52) \times (3/51) = 1/221 ( 4/52 ) × ( 3/51 ) = 1/221
Permutation n P r = n ! ( n − r ) ! _n P_r = \frac{n!}{(n-r)!} n P r = ( n − r )! n ! "Order matters", ranking, positions, codes Ways to assign President, VP, Secretary from 10: 10 P 3 = 720 _{10}P_3 = 720 10 P 3 = 720
Combination n C r = n ! r ! ( n − r ) ! _n C_r = \frac{n!}{r!(n-r)!} n C r = r ! ( n − r )! n ! "Order does not matter", group, committee Ways to choose 4 books from 9: 9 C 4 = 126 _9 C_4 = 126 9 C 4 = 126