Representing data flashcards

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Mathematical model (statistics)
A simplified description of a real-world situation, based on assumptions, used to make predictions.
Stages of the statistical modelling cycle
Real-world problem → assumptions and model → predictions → collect data → compare data with predictions → refine the model and repeat.
Class boundaries
The values where one class ends and the next begins, e.g. 19.5 and 24.5 for lengths 20–24 recorded to the nearest unit.
Histogram
A diagram for continuous grouped data: the bars touch and the area of each bar is proportional to the class frequency.
Modal class (unequal widths)
The class with the largest frequency density, not necessarily the largest frequency.
Interquartile range (IQR)
Q₃ − Q₁: the spread of the middle half of the data; not affected by extreme values.
Outlier
An extreme value outside limits stated in the question, e.g. beyond Q₁ − 1.5 × IQR or Q₃ + 1.5 × IQR.
Positive skew
Long tail to the right: Q₃ − Q₂ > Q₂ − Q₁ and mode < median < mean.
Negative skew
Long tail to the left: Q₃ − Q₂ < Q₂ − Q₁ and mean < median < mode.
Frequency density
frequency density = frequency ÷ class width. (No — learn it)
Area of a histogram bar
area = k × frequency (k = 1 when the vertical axis is frequency density). (No — learn it)
Positions of Q₁, Q₂, Q₃ for n raw values
n/4, n/2, 3n/4. Whole number r: mean of the rth and (r + 1)th values; otherwise round up. (No — learn it)
Outlier fences (common rule)
Q₁ − 1.5 × (Q₃ − Q₁) and Q₃ + 1.5 × (Q₃ − Q₁); the question states the multiplier. (No — learn it)
A skewness coefficient a question may give
3(mean − median) ÷ standard deviation: positive → positive skew, negative → negative skew. (No — learn it)
What the area of each histogram bar represents
The frequency of that class (area is proportional to frequency).
Stem-and-leaf diagram: what must be included
Ordered leaves on each stem and a key, e.g. 3 | 4 means 34.
Reading the left side of a back-to-back stem-and-leaf diagram
Leaves are read outwards from the stem: smallest next to the stem. The key says how each side reads.
Box plot: where the whiskers end
At the smallest and largest values that are not outliers; outliers are plotted as crosses.
Box plot of a positively skewed data set
Median nearer Q₁; the right half of the box and the right whisker are longer.
Two advantages of a statistical model
Quick and cheap compared with real experiments; gives predictions for planning; easy to change; helps understanding.
Two limitations of a statistical model
Simplifies reality so predictions are approximate; assumptions may be wrong; may hold only for a restricted range.
Why histogram bars must have frequency density as height when widths differ
Otherwise a wide class would look too important; area, not height, must show frequency.
Assumption when estimating a frequency for part of a class
The values are spread evenly across the class.
Which average and spread to use for skewed data or data with outliers
Median and IQR, because they are hardly affected by extreme values.
What a full comparison of two distributions contains
A measure of location and a measure of spread (plus skew or outliers if relevant), each interpreted in context.
Should an outlier be removed?
Only if there is reason to believe it is an error; a genuine unusual value stays in the data.
Class 2.5 ≤ x < 4 has frequency 21. Frequency density?
21 ÷ 1.5 = 14.
Classes 10–15 (fd 3.6), 15–20 (frequency 25), 20–30 (fd 3). Estimate the number between 12 and 25.
3 × 3.6 + 25 + 5 × 3 = 50.8 ≈ 51.
Bar 2 cm wide, 6 cm tall shows 15 values (class width 5). A class of width 15 has 9 values: bar size?
Width 15 × 0.4 = 6 cm; area 9 × 0.8 = 7.2 cm², so height 1.2 cm.
19 ordered values: 5th = 35, 10th = 42, 15th = 53; largest values 64 and 91. Outliers (1.5 × IQR rule)?
IQR 18, upper fence 53 + 27 = 80, so 91 is an outlier; the whisker ends at 64.
n = 27 raw values. Which values give Q₁, Q₂, Q₃?
6.75 → 7th; 13.5 → 14th; 20.25 → 21st.
n = 16 raw values. How are Q₁, Q₂, Q₃ found?
Positions 4, 8, 12 are whole numbers: mean of 4th and 5th, 8th and 9th, 12th and 13th values.
Mean 186, median 172, sd 42. Evaluate 3(mean − median)/sd and interpret.
3 × 14 ÷ 42 = 1: positive skew.
Mean 23.4, sd 6.5; outliers are more than 2 sd from the mean. Is 4 an outlier?
Lower limit 23.4 − 13 = 10.4; 4 < 10.4, so yes.
Class 20–24 for ages in completed years versus lengths to the nearest cm
Ages: boundaries 20 and 25. Lengths: 19.5 and 24.5. Both have width 5.
Skew named from the peak
Wrong: skew is named after the long tail. Peak on the left, tail to the right = positive skew.
Histogram scale in cm: assuming height = frequency
Find the cm-per-fd (or cm²-per-value) scale from a known bar first, then convert.
Comparing reaction or race times: higher median = better?
No — the context decides: a lower time is better.

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