← Praxis Core: Statistics & Geometry

Praxis Core Teaching Exam Study Guide

Key concepts, definitions, and exam tips organized by topic.

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Praxis Core: Statistics & Geometry — Study Guide


Overview

The Praxis Core Math exam tests foundational skills in statistics and geometry, including interpreting data, calculating measures of central tendency, computing area and volume, and applying geometric relationships. This guide covers all major concepts you need to master, organized by topic with key formulas, definitions, and exam tips. Use the Quick Review Checklist at the end to confirm your readiness.


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Measures of Central Tendency


Core Concepts

The three primary measures of central tendency describe where data clusters.


  • • Mean — The arithmetic average. Add all values, then divide by the count.

    • - Example: {4, 7, 7, 9, 13} → 40 ÷ 5 = 8

  • • Median — The middle value of an ordered dataset.

    • - Odd number of values: the single middle value

    - Even number of values: the average of the two middle values

    - Example: {2, 3, 6, 8, 15} → median = 6

  • • Mode — The value that appears most frequently. A dataset can have one mode, multiple modes, or no mode.

    • - Example: {5, 3, 9, 3, 7, 5, 3} → mode = 3 (appears 3 times)

  • • Range — A measure of spread, not central tendency. Calculated as maximum − minimum.

    • - Example: {7, 12, 28, 33, 45} → 45 − 7 = 38


    Choosing the Right Measure

    | Situation | Best Measure |

    |---|---|

    | Symmetric data, no outliers | Mean |

    | Outliers present | Median |

    | Categorical or repeated values | Mode |


    Key Terms

  • • Outlier — An extreme value that significantly differs from the rest of the data
  • • Resistant measure — A statistic not heavily affected by outliers (the median is resistant; the mean is not)

    • Watch Out For

    > ⚠️ Always sort the data first before finding the median. A common mistake is picking the middle position from an unsorted list.

    >

    > ⚠️ When an outlier is present, the mean is pulled toward the extreme value. The median remains stable — making it the better descriptor of "typical" in those cases.

    >

    > ⚠️ Range is NOT a measure of central tendency — it measures spread. Don't confuse the two.


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    Data Interpretation


    Types of Graphs

    | Graph Type | Best Used For | Key Feature |

    |---|---|---|

    | Bar Graph | Comparing discrete categories | Gaps between bars |

    | Histogram | Showing continuous data distributions | No gaps between bars |

    | Pie Chart | Showing parts of a whole (percentages) | Sections sum to 100% |

    | Line Graph | Showing change over time | Slope = rate of change |

    | Stem-and-Leaf Plot | Showing actual data values + distribution | Preserves original data |

    | Scatterplot | Showing relationships between two variables | Points show correlation |


    Key Calculations

  • • Percent increase/decrease:

    • $$\frac{\text{New} - \text{Original}}{\text{Original}} \times 100$$

    Example: Bar goes from 25 to 40 → (15 ÷ 25) × 100 = 60% increase


  • • Finding a part from a percentage:

    • Multiply total × decimal form of percent

    Example: 200 students × 0.30 = 60 students


    Correlation in Scatterplots

  • • Positive correlation — Points trend upward left to right (as x increases, y increases)
  • • Negative correlation — Points trend downward left to right (as x increases, y decreases)
  • • No correlation — Points show no clear pattern

    • Key Terms

  • • Distribution — How data values are spread across a range
  • • Rate of change — How quickly a value changes (represented by slope on a line graph)
  • • Continuous data — Data that can take any value within a range (shown in histograms)
  • • Discrete data — Data in separate, countable categories (shown in bar graphs)

    • Watch Out For

    > ⚠️ Histogram ≠ Bar Graph. The key visual difference is that histogram bars touch (no gaps); bar graph bars have spaces. Histograms show intervals of continuous data.

    >

    > ⚠️ Steeper slope on a line graph = faster rate of change, not necessarily a higher value.

    >

    > ⚠️ Correlation does not imply causation. A scatterplot showing a relationship does not mean one variable causes the other.


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    Probability & Statistics Concepts


    Basic Probability

    $$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

    Example: 3 blue out of 12 marbles → P(blue) = 3/12 = 1/4


    Key Probability Rules

  • • Mutually Exclusive Events — Two events that cannot occur at the same time.

    • - P(A or B) = P(A) + P(B)

    - Example: Rolling a 2 or a 5 on a single die roll

  • • Independent Events — The outcome of one event does not affect the other.

    • - P(A and B) = P(A) × P(B)

    - Example: Flipping heads twice → (1/2) × (1/2) = 1/4


    Distribution Shape

  • • Skewed Right (Positively Skewed)

    • - Long tail extends to the right

    - Most values cluster on the left

    - A few very high values pull the mean above the median

    - Mean > Median > Mode

  • • Skewed Left (Negatively Skewed)

    • - Long tail extends to the left

    - Mean < Median < Mode

  • • Symmetric (Normal)

    • - Mean ≈ Median ≈ Mode


    Key Terms

  • • Independent events — Events where the outcome of one does not influence the other
  • • Mutually exclusive — Events that cannot happen simultaneously
  • • Skewed distribution — A dataset where data is not evenly spread around the center

    • Watch Out For

    > ⚠️ In a skewed distribution, the mean is pulled toward the tail, while the median stays closer to the center. Always think: the tail points to the skew direction, and pulls the mean that way.

    >

    > ⚠️ Multiplying probabilities only works for independent events. If events are dependent, you must adjust the second probability.

    >

    > ⚠️ Don't forget to simplify fractions when expressing probability (3/12 = 1/4).


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    Geometry: Area & Perimeter


    Essential Formulas


    | Shape | Perimeter | Area |

    |---|---|---|

    | Rectangle | P = 2l + 2w | A = l × w |

    | Triangle | P = a + b + c | A = ½ × b × h |

    | Trapezoid | P = sum of all sides | A = ½(b₁ + b₂) × h |

    | Circle | C = π × d or 2πr | A = π × r² |


    Worked Examples

  • • Rectangle: l = 12, w = 7 → A = 12 × 7 = 84 cm²
  • • Triangle: b = 10, h = 6 → A = ½ × 10 × 6 = 30 in²
  • • Circle circumference: d = 10 → C = 3.14 × 10 = 31.4 cm
  • • Circle area: r = 5 → A = 3.14 × 25 = 78.5 m²
  • • Triangle perimeter: 5 + 8 + 11 = 24 cm
  • • Trapezoid: b₁ = 6, b₂ = 10, h = 4 → A = ½(16) × 4 = 32 units²

    • Key Terms

  • • Base (b) — A side of a shape used as a reference for area calculations
  • • Height (h) — The perpendicular distance from base to the opposite side/vertex
  • • Radius (r) — Distance from center to edge of a circle
  • • Diameter (d) — Distance across a circle through its center; d = 2r

    • Watch Out For

    > ⚠️ The height of a triangle must be perpendicular to the base — it is not always a side of the triangle.

    >

    > ⚠️ Circle formulas use radius, not diameter. If given the diameter, divide by 2 first before using A = πr².

    >

    > ⚠️ Trapezoid area requires BOTH bases (b₁ and b₂), not just one.

    >

    > ⚠️ Area uses square units (cm², m²); circumference/perimeter uses linear units (cm, m).


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    Geometry: Volume & 3D Shapes


    Essential Formulas


    | Shape | Formula |

    |---|---|

    | Rectangular Prism | V = l × w × h |

    | Cylinder | V = π × r² × h |

    | Cube | V = s³ |


    Worked Examples

  • • Rectangular Prism: 6 × 4 × 3 = 72 cm³
  • • Cylinder: r = 3, h = 10 → V = 3.14 × 9 × 10 = 282.6 units³

    • Key Terms

  • • Volume — The amount of 3-dimensional space inside a figure (measured in cubic units)
  • • Prism — A 3D shape with two identical, parallel bases connected by rectangular faces

    • Watch Out For

    > ⚠️ Volume is always expressed in cubic units (cm³, m³, in³).

    >

    > ⚠️ For a cylinder, the r² applies only to the radius — square the radius before multiplying by π and h.


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    Geometry: Angles & Shapes


    Angle Relationships

    | Relationship | Definition | Sum |

    |---|---|---|

    | Complementary angles | Two angles that form a right angle | 90° |

    | Supplementary angles | Two angles that form a straight line | 180° |

    | Vertical angles | Opposite angles formed by two intersecting lines | Equal to each other |


    Example: One supplementary angle = 65° → other angle = 180° − 65° = 115°


    Interior Angle Sums

    | Shape | Sum of Interior Angles |

    |---|---|

    | Triangle | 180° |

    | Quadrilateral | 360° |

    | Pentagon | 540° |

    | Hexagon | 720° |

    | Any polygon | (n − 2) × 180° |


    The Pythagorean Theorem

    For any right triangle with legs a and b, and hypotenuse c:

    $$a^2 + b^2 = c^2$$

    Example: Legs = 6 and 8 → 36 + 64 = 100 → √100 = 10


    Common Pythagorean triples to memorize:

  • • 3 – 4 – 5
  • • 5 – 12 – 13
  • • 6 – 8 – 10 (multiple of 3-4-5)
  • • 8 – 15 – 17

    • Key Terms

  • • Hypotenuse — The longest side of a right triangle, always opposite the right angle
  • • Legs — The two shorter sides of a right triangle that form the right angle
  • • Quadrilateral — Any four-sided polygon (includes squares, rectangles, trapezoids, parallelograms)

    • Watch Out For

    > ⚠️ The Pythagorean theorem only applies to RIGHT triangles. If the triangle is not a right triangle, this formula cannot be used.

    >

    > ⚠️ The hypotenuse is always the side opposite the 90° angle — it is always c, never a or b.

    >

    > ⚠️ Supplementary ≠ Complementary. Supplementary = 180°; Complementary = 90°. A helpful memory trick: Complementary = Corner (right angle = 90°).

    >

    > ⚠️ All quadrilaterals have interior angles summing to 360° — this includes irregular shapes, not just squares and rectangles.


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    Quick Review Checklist


    Use this checklist before your exam. Check off each item as you feel confident:


    Measures of Central Tendency

  • • [ ] Calculate mean, median, and mode from a dataset
  • • [ ] Sort data before finding the median
  • • [ ] Find the median of an even-numbered dataset (average the two middle values)
  • • [ ] Calculate the range (max − min)
  • • [ ] Know when to use the median vs. mean (outliers present → use median)

    • Data Interpretation

  • • [ ] Calculate percent increase/decrease using the correct formula
  • • [ ] Find a partial count from a percentage (total × decimal)
  • • [ ] Distinguish a histogram (no gaps) from a bar graph (gaps)
  • • [ ] Identify positive/negative/no correlation from a scatterplot
  • • [ ] Interpret slope on a line graph as rate of change

    • Probability & Statistics

  • • [ ] Calculate basic probability (favorable ÷ total)
  • • [ ] Apply multiplication rule for independent events
  • • [ ] Identify mutually exclusive events
  • • [ ] Describe a right-skewed distribution (mean > median, tail on right)

    • Area & Perimeter

  • • [ ] Apply area formulas: rectangle, triangle, trapezoid, circle
  • • [ ] Apply perimeter/circumference formulas for all shapes
  • • [ ] Use radius (not diameter) in circle formulas

    • Volume

  • • [ ] Calculate volume of a rectangular prism (l × w × h)
  • • [ ] Apply cylinder volume formula (πr²h)
  • • [ ] Label answers in cubic units

    • Angles & Shapes

  • • [ ] Know supplementary (180°) vs. complementary (90°) angles
  • • [ ] State the interior angle sums: triangle = 180°, quadrilateral = 360°
  • • [ ] Apply the Pythagorean theorem to find missing sides in right triangles
  • • [ ] Recall common Pythagorean triples (3-4-5, 5-12-13, 6-8-10)

    • ---

    Good luck on the Praxis Core! Focus on formula fluency and read each problem carefully to identify what's being asked before calculating.

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