MAT-144 · Mathematical Reasoning Topic 05 · Statistics
Topic 05 · Cheat Sheet

Study card

The formulas, the moves, and the traps for Topic 5, in one printable page.

Key formulas

The five descriptive measures, the bell curve's three-band rule, and the margin-of-error formula that powers every poll headline.

Mean (arithmetic average)
mean = sum of valuesnumber of values
Add everything, divide by how many. Sensitive to outliers — one huge value can pull the mean far off-center. Excel: =AVERAGE(A2:A10).
Median (the middle value)
Sort first. Then:
odd count  →  the single middle value
even count  →  average of the two middle values
Robust to outliers — that's why home prices and salaries are usually reported as medians. Excel: =MEDIAN(A2:A10).
Mode (most frequent)
mode = the value that appears most often
If every value appears once, there is no mode. Ties are allowed (a set can be bimodal). Excel: =MODE.SNGL(A2:A10).
Range
range = max − min
The simplest measure of spread — ignores everything in the middle. Two data sets can have the same mean and very different range. Excel: =MAX(A2:A10)-MIN(A2:A10).
Sample standard deviation
s = √( Σ(x − x̄)2n − 1 )
Subtract the mean from each value, square, sum, divide by n − 1, take the square root. ALEKS expects sample SD (denominator n − 1), not population SD. Excel: =STDEV.S(A2:A10).
Empirical rule (68 · 95 · 99.7)
≈ 68% of values fall within 1 SD of the mean
≈ 95% of values fall within 2 SDs of the mean
≈ 99.7% of values fall within 3 SDs of the mean
Only works when the data is approximately normal (the bell curve). For one-sided questions (above or below), halve the symmetric percentage and add/subtract from 50%.
Margin of error (95% confidence, proportion)
MoE ≈ 1√n   (rough rule of thumb)
A sample of n = 1,000 gives MoE ≈ ±3% — the number behind every “45% support, ±3%” headline. To halve the margin, you have to quadruple the sample size. The full formula uses 1.96 √(p(1−p)/n); the 1/√n rule is what you'd quote in a hurry.

Common mistakes

The traps that cost real points on the Topic 5 review.

  1. Forgot to sort before taking the median. The median is the middle of the sorted data set, not the middle of whatever order ALEKS handed you. Always sort first.
  2. Even-count median taken as a single value. With ten values, the median is the average of the 5th and 6th values, not just “the one in the middle.”
  3. Claiming a mode when every value appears once. If no value repeats, the answer is “no mode,” not the smallest or the median.
  4. Used n in standard deviation when ALEKS wants sample SD. Sample SD divides by n − 1. Excel's STDEV.S matches; STDEV.P divides by n and gives the wrong answer for ALEKS-style problems.
  5. Forgot to square the deviations. The deviations (x − x̄) always sum to zero — that's why you square first. Skipping the square gives 0 every time.
  6. Empirical rule applied to non-normal data. 68/95/99.7 only works for approximately normal (bell-shaped) distributions. Don't quote it for skewed data like incomes or home prices.
  7. Confused “within 2 SD” with “above 2 SD.” 95% is within 2 SDs. The two tails outside that range hold the remaining 5%, split as 2.5% each side.
  8. Misread the data display. A bar chart's height is a frequency. A histogram's bars touch (continuous). A pie's slice is a percentage of the whole. Read the axis labels before reading the bars.
  9. Treated descriptive numbers as inferential claims. “The mean of these 10 weights is 142 lb” is descriptive. “The mean weight of all 11th graders is 142 lb” is inferential and needs a margin of error.
  10. Margin of error read as a percentage of the percentage. “±3%” on a 45% poll means the true value is between 42% and 48%, not between 43.65% and 46.35%. The ±3 is in percentage points, not relative percent.

Quick reference

Visual cheats and Excel patterns for the descriptive-statistics half of MAT-144.

Pick the right center
Mean — use when the data is roughly symmetric and outliers are rare (test scores, heights).
Median — use when the data is skewed or has outliers (incomes, home prices, response times).
Mode — use for categorical data, or when you want the most-common single value (shoe size, exam grade).
Reading data displays
Bar chart — categorical data; bars have gaps; height = frequency.
Histogram — numeric data binned into intervals; bars touch; height = frequency in that bin.
Pie chart — parts of a whole; slice = percentage of the total. Use only when the categories add up to 100%.
Line chart — trend over time; x-axis is ordered (years, months).
Box plot — five-number summary (min, Q1, median, Q3, max); the box is the middle 50%.
Empirical-rule cheat lines
between mean − 1s and mean + 1s  →  68%
between mean − 2s and mean + 2s  →  95%
between mean − 3s and mean + 3s  →  99.7%
below mean − 2s  →  2.5%  (half of the 5% in the tails)
Excel for statistics
=AVERAGE(A2:A10)  mean
=MEDIAN(A2:A10)  median
=MODE.SNGL(A2:A10)  single mode
=MAX(A2:A10)-MIN(A2:A10)  range
=STDEV.S(A2:A10)  sample standard deviation
=COUNT(A2:A10)  n (number of values)
=NORM.DIST(x, mean, sd, TRUE)  area under bell curve up to x
Descriptive vs inferential
Descriptive — summarizes the data set you have. Mean / median / mode / range / SD. Claim is about this data.
Inferential — uses a sample to make a claim about a larger population. Polls, surveys, A/B tests. Claim has a margin of error.
Topic 5 overview Glossary
MAT-144 · TOPIC 05 · CHEAT SHEET