Triola Elementary Statistics · 11th Edition
Chapter 3: Statistics for Describing Data
Every key formula from Triola Elementary Statistics Chapter 3, in one searchable page. Click a card to study it — worked examples included.
3.2 · MEASURES OF CENTER
Mean
x̄ = ∑xn (sample) · μ = ∑xN (population)
Sample statistics use English letters (x̄); population parameters use Greek (μ).
Median
Odd count: the exact middle value (after sorting). Even count: average the two middle values. Resistant — outliers barely move it.
Mode
The value that occurs most often. Can be bimodal or “no mode.” Rarely used with numerical data.
Midrange
midrange = max + min2
Quick but too sensitive to extremes — rarely used.
3.3 · MEASURES OF VARIATION
Range
range = max − min
Sample standard deviation
s = √∑(x − x̄)²n − 1
s = √n∑x² − (∑x)²n(n − 1) (shortcut — what calculators use)
Population standard deviation
σ = √∑(x − μ)²N
Divide by N, not n − 1. Variance s²/σ² = the square of the SD (units: squared!).
Range rule of thumb
s ≈ range/4 · usual = mean ± 2s
IQ: μ = 100, σ = 15 ⇒ usual = 70 to 130; IQ 135 ⇒ unusual.
3.4 · PERCENTILES
Percentile of a value x
percentile = # values < xtotal # values × 100
Finding Pk (the kth percentile)
L = k100 · n
L not whole ⇒ round UP to the next whole number, take that value.
L whole ⇒ average the Lth value and the next one.
Q1 = P25, Q2 = P50 (= median), Q3 = P75.
L whole ⇒ average the Lth value and the next one.
Q1 = P25, Q2 = P50 (= median), Q3 = P75.
WORKED EXAMPLE
n = 35, find P90: L = 31.5 ⇒ round up ⇒ 32nd value.
Interquartile range
IQR = Q3 − Q1
Outlier fences: below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.
Empirical (68-95-99.7) rule — bell-shaped data only
68% within 1 SD of the mean · 95% within 2 SDs · 99.7% within 3 SDs
IQ: about 95% of scores fall between 70 and 130.
Chebyshev’s theorem — ANY data set
At least 1 − 1/k² of values lie within k SDs of the mean (k > 1)
75% within 2 SDs · 89% within 3 SDs. Weaker than the empirical rule, but always true.
WORKED EXAMPLE
IQ: μ = 100, σ = 15. Chebyshev (k = 2):
at least 75% of scores are within 2 SDs ⇒ between 70 and 130.
at least 75% of scores are within 2 SDs ⇒ between 70 and 130.
Standardized value
z = x − x̄s · z = x − μσ
The number of standard deviations x is above (+) or below (−) the mean. Round to 2 decimals.
Usual vs unusual
Ordinary: −2 ≤ z ≤ 2 · Unusual: z < −2 or z > 2
WORKED EXAMPLE
IQ 135 (μ = 100, σ = 15):
z = (135 − 100)/15 = 2.33 > 2 ⇒ unusual.
z = (135 − 100)/15 = 2.33 > 2 ⇒ unusual.
z-scores also let you compare values from different data sets (height in inches vs weight in pounds).
3.4 · 5-NUMBER SUMMARY
5-number summary
min, Q1, median (Q2), Q3, max
Movie budgets: 4.5, 35, 68, 113, 225 ($ millions).
Boxplot (box-and-whisker)
1. Find the 5-number summary. 2. Draw a scale covering min to max. 3. Draw a box from Q1 to Q3 with a line at the median. 4. Draw whiskers from the box out to the min and max.
KEY NOTES
- Boxplots show spread and center at a glance — great for comparing groups.
- Extreme outliers in boxplots get objective cutoff criteria; but remember outliers can distort the mean and SD.
WATCH OUT!
Mistakes that cost points
- Sample SD divides by n − 1, not n; population divides by N
- Variance units are squared — the SD is in the original units
- Empirical rule only for bell-shaped data; Chebyshev works for any data
- Locator L: round UP if not whole; average if whole
- z-score denominators: sample ⇒ s, population ⇒ σ
- The mean chases outliers; the median resists them