Triola Elementary Statistics · 11th Edition
Chapter 2: Summarizing & Graphing Data
Every key formula from Triola Elementary Statistics Chapter 2, in one searchable page. Click a card to study it — worked examples included.
2.2 · FREQUENCY DISTRIBUTIONS
Building one by hand
1. Use 5–20 classes. 2. Class width = (max − min) / #classes — round UP to a convenient number. 3. First lower limit = min (or below it). 4. Add the width for the remaining limits. 5. Tally every value into exactly one class — include classes with frequency 0.
WORKED EXAMPLE
Pulse rates: max 124, min 60, 7 classes.
124 − 607 = 9.1428571 ⇒ round UP to 10.
First limits: 60, 70, 80, 90, 100, 110, 120.
124 − 607 = 9.1428571 ⇒ round UP to 10.
First limits: 60, 70, 80, 90, 100, 110, 120.
Key parts
Boundaries (59.5, 69.5, …) leave no gaps. Midpoint = (lower + upper)/2.
Width = lower-limit difference = 10, not 9!
Width = lower-limit difference = 10, not 9!
2.2 · RELATIVE & CUMUL. FREQ.
Relative frequency
rel. freq. = class frequencysum of all frequencies
12 of 40 ⇒ 12/40 = 0.3 = 30%. The sum must be ≈ 1 (or 100%) — small rounding errors OK.
Cumulative frequency
Running total of this class + all previous classes
Frequencies 12, 14, 11 ⇒ cumulative 12, 26, 37, …
KEY NOTES
- Relative frequencies let you compare data sets of different sizes.
- Ogives plot cumulative frequencies — use them to read off “how many values are below X?”
2.3 · HISTOGRAMS
Bars touch (quantitative data on the scale). Read the histogram with CVDOT: Center, Variation, Distribution shape, Outliers, change over Time.
Normal (bell) distribution
Frequencies rise to a max, then fall · symmetric (left ≈ mirror of right)
Many later methods require an approximately normal distribution — a histogram is the quick check.
Other shapes
Uniform: all bars about equal. Skewed right: tail stretches right. Skewed left: tail stretches left.
The pulse-rate histogram is heavier on the left — the tall bar at ~125 is an outlier.
The pulse-rate histogram is heavier on the left — the tall bar at ~125 is an outlier.
2.4 · STATISTICAL GRAPHICS
Frequency polygon
Plot points above class midpoints, connect with line segments. Great for comparing two data sets (females vs males) on the same axes — use the relative version when sample sizes differ.
Ogive (“oh-jive”)
Cumulative frequencies vs class boundaries — read off how many values fall below a given value.
Dotplot
Each data value = one dot above a scale. Shows every value, quick to read.
Stemplot (stem-and-leaf)
Separates each value into a stem and leaves — shows the distribution while keeping every value; also sorts the data for you.
2.4 · CATEGORICAL GRAPHS
Pareto chart
Bar graph for qualitative data with bars in descending order of frequency — the “vital few” stand out first.
Pie chart
central angle = rel. freq. × 360°
Slices of a circle for qualitative data. 61% of workers found jobs through networking ⇒ 0.61 × 360° = 220° slice.
KEY NOTES
- Pie charts look friendly; Pareto charts show relative sizes more clearly.
- Both are for categorical data — never use them for quantitative distributions.
2.4 · SCATTER & TIME SERIES
Scatterplot
Plot of paired (x, y) quantitative data: x on the horizontal axis, y on the vertical.
The pattern of points hints at a relationship — cricket chirps rise with temperature ⇒ crickets as thermometers!
The pattern of points hints at a relationship — cricket chirps rise with temperature ⇒ crickets as thermometers!
Watch for clusters
Penny weights vs year show two clusters with a gap (pre-1983 vs post-1983 copper content). Lumping them hides the real story — examine groups separately.
Time-series graph
Quantitative data collected at different points in time — reveals trends and change.
WATCH OUT!
Mistakes that cost points
- Class width: round UP, and use consecutive lower limits (not upper − lower)
- Classes must not overlap; every value goes in exactly one class
- Relative frequencies must sum to ≈ 1 (or 100%)
- Histogram: bars touch for quantitative data; bar gaps are for categorical bar graphs
- Ogive uses boundaries + cumulative frequencies; polygon uses midpoints
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