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Flashcards · AP® Statistics · Unit 1

Unit 1: Exploring One-Variable Data and Collecting Data flashcards

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1.1 Introducing Statistics: What Can We Learn from Data?

What makes a question a statistical investigative question?
It can only be answered with data that vary from one individual to another, using a sample to learn about a whole population. "What time does the bus leave?" has one fixed answer, so it is not statistical.
What is the difference between N and n?
Capital N is the population size; lower-case n is the sample size, with n less than N. A club asking 120 of its 900 members has N = 900 and n = 120.
What are a population and a sample?
The population is all the items or individuals of interest. The sample is the subset the data actually come from. Do not call the sample "the population".
Why do studies use a sample instead of the whole population?
The population is often too large, or too hard, costly or destructive to measure in full.
What does writing "in context" mean?
Saying what each part of the study and each number matches in the real situation: 42 of the 120 club members asked (35%) said they want a summer trip, not just "35%".
Is a sample result exactly the population value?
No. It describes the sample; another sample would probably give a slightly different value.
Why must an investigative question be fixed before analysis?
Changing the question after seeing results makes the evidence look stronger than it is. Vague words like "good" or "like" also need a defined, measurable meaning.

1.2 Variables

What is an observational unit?
The item or individual one piece of data is about: a person, household, tree, day, photo or review. Ask "one row describes one what?"; units are not always people.
How do categorical and quantitative variables differ?
Categorical values are category names or group labels, where an average is meaningless. Quantitative values are measured or counted amounts, usually with units, where an average makes sense.
Is a number always a quantitative variable?
No. Numbers used only as labels (ID, postcode, jersey or phone number) are categorical.
How do discrete and continuous quantitative variables differ?
Discrete values are countable (0, 1, 2, …), even with no upper limit, such as goals scored. Continuous values can be any value in an interval, such as time in seconds; rounding a measurement does not make it discrete.
What is the difference between a parameter and a statistic?
A parameter summarises a variable for the whole population and is usually unknown. A statistic summarises a sample and estimates the parameter. 120 oaks of all 400 trees (0.3) is a parameter; 7 of 20 sampled (0.35) is a statistic.
Does the type of summary decide whether it is a parameter or a statistic?
No. A mean, median or proportion can be either; only whether it comes from the population or a sample matters.

1.3 Tabular Representation and Summary Statistics for One Categorical Variable

What do frequency and relative frequency tables show?
A frequency table gives the count in each category (counts add to n). A relative frequency table gives the proportion in each category: count ÷ n, between 0 and 1, adding to 1.
How do you get a count from a proportion?
Count = proportion × n, so you must know n. 0.15 of 320 trains late is 0.15 × 320 = 48 trains.
What is the difference between "a majority" and "the most common category"?
A majority needs a proportion greater than 0.5. The most common category has the largest count or proportion, which can be below 0.5.
How do you convert a part-to-rest ratio to a proportion?
For a : b with no other categories, proportion = a ÷ (a + b). Yes to No of 2 : 3 gives 2 ÷ 5 = 0.40 saying yes. 9 : 16 is not 9 out of 16.
Why compare proportions rather than counts between groups?
Groups of different sizes give counts that are not comparable; proportions put them on the same scale.
How should a categorical variable be summarised?
With counts, proportions and the most common category, never a mean or median. Each individual must be in exactly one category.

1.4 Graphical Representations for One Categorical Variable

What do bar charts and pie charts show for a categorical variable?
A bar chart has one bar per category, with height equal to frequency or relative frequency. A pie chart shows each category's relative frequency as its share of the whole circle.
How do you find a pie chart slice angle?
Angle = relative frequency × 360°. 0.15 gives 54°; 30% is 108°, not 30°. All angles add to 360°.
How do you find a count from a pie chart slice?
Count = (angle ÷ 360°) × n. A 90° slice of 64 people is 16 people.
When should you not use a pie chart?
For "tick all that apply" data, where parts add to more than the whole; use a bar chart. Every individual must be in exactly one category for a pie chart.
Can a bar chart be described as skewed?
No. Categories have no number-line order, so describe the most and least common categories instead.
What are the rules for drawing a bar chart?
Bars are separate and equal in width, the frequency axis starts at 0 (otherwise differences are exaggerated), and bars are ordered by size or by a natural order.
Do frequency and relative frequency bar charts look different?
They have the same shape; only the axis scale differs.

1.5 Graphical Representations for One Quantitative Variable

How do dotplots, stem-and-leaf plots and histograms differ?
Dotplots and stem-and-leaf plots keep every value. A histogram groups values into ordered, equal-width bins and keeps only the count in each bin, so exact values cannot be read from it.
What must a stem-and-leaf plot include?
Every stem in the range, even empty ones; single-digit leaves in order; and a key such as 7 | 3 = 73 kg. The row 8 | 0 4 4 shows 80, 84 and 84 kg.
How is a histogram drawn correctly?
Equal-width bins covering every value, touching bars, labelled axes with units, vertical scale starting at 0, and a stated boundary rule, such as "a value on a boundary goes in the bin on its right", so 30 goes in the 30-to-40 bin.
How do you find a relative frequency from a histogram bar?
Relative frequency = frequency ÷ n. A bin with 6 of 30 values has 0.20. A bar of height 0.15 in a histogram of 60 values holds 9 values.
Why might a histogram's appearance change with bin width?
Different bin widths group values differently and can change the apparent shape, so try more than one width and state the one you use.
What are split stems?
Each stem is written twice, once for leaves 0–4 and once for leaves 5–9, to spread out a crowded stem-and-leaf plot.

1.6 Descriptions for One Quantitative Variable Distributions

What four parts make a full description of a quantitative distribution?
Shape, centre, variability and unusual features, all in context with the variable, individuals and units.
How do you name the direction of skew?
By the longer tail, not the peak. A peak on the left with a long tail toward larger values is skewed right; a long tail toward smaller values is skewed left.
What words describe the number of peaks?
Unimodal (one main peak), bimodal (two clear peaks) or approximately uniform (about equal frequencies, no clear peak). Frequencies 2, 9, 4, 1, 8, 3 are bimodal.
What are gaps, clusters and possible outliers?
A gap is a stretch of the scale with no data. A cluster is a group of values close together, set apart by gaps. A possible outlier is a value unusually far from the rest; a large group beyond a gap is a cluster, not an outlier.
How do you estimate the median from a histogram?
Name the bin that holds the middle value of the ordered data; you cannot claim an exact median from a histogram.
How do you use a graph to judge a claim?
Quote counts or percentages read from the graph and compare them with the claim, rather than judging by appearance.

1.7 Summary Statistics for One Quantitative Variable

How do you calculate the mean and median?
Mean x̄ = Σxᵢ / n, using every value. Median: order the data and take the middle value (n odd) or the mean of the two middle values (n even).
How do you find Q1, Q3 and the IQR?
Split the ordered data at the median (leave it out of both halves when n is odd); Q1 and Q3 are the medians of the halves. IQR = Q3 − Q1. Data 4, 7, 7, 9, 13 kg: Q1 = 5.5, Q3 = 11, IQR = 5.5 kg.
What is the sample standard deviation?
s = √[Σ(xᵢ − x̄)² / (n − 1)], a typical distance of values from the mean (Sx on a calculator, not σx). The variance is s², in squared units.
Which summary statistics are resistant to outliers?
Median and IQR are resistant. Mean, range and standard deviation are not, so prefer median and IQR for skewed data or data with outliers.
What is the 1.5 × IQR outlier rule?
A value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is a potential outlier. Q1 = 30 s, Q3 = 42 s: upper fence 42 + 18 = 60 s, so 61 s is flagged. Investigate; do not delete unless it is an error.
What is the 2s outlier rule?
A value below x̄ − 2s or above x̄ + 2s is a potential outlier. It uses non-resistant statistics.
How do linear changes to data affect summary statistics?
Multiply by k (k > 0): centre, position, range, IQR and s all multiply by k; variance by k². Add c: centre and position increase by c; range, IQR and s are unchanged. Mean 12 cm, s = 3 cm becomes 120 mm and 30 mm.
What is a percentile?
The p-th percentile has p% of the values at or below it. Q1 is the 25th percentile and Q3 the 75th.

1.8 Graphical Representations of Summary Statistics for One Quantitative Variable

What is the five-number summary?
Minimum, Q1, median, Q3, maximum. A boxplot draws it on a labelled number line.
How is a boxplot drawn?
The box runs from Q1 to Q3 with a line at the median (not the mean). Whiskers run to the most extreme values that are not outliers; each outlier gets its own symbol. Fences are Q1 − 1.5 × IQR and Q3 + 1.5 × IQR.
Does a longer section of a boxplot contain more data?
No. Each of the four sections holds about 25% of the data; a longer section means more spread.
How do the mean and median indicate skew?
The mean is pulled toward a long tail; the median is not. Skewed right: mean usually greater than median. Skewed left: mean usually less. Mean 41 and median 52 suggests skewed left.
What can a boxplot not show?
n, gaps, clusters, the number of peaks, or the mean (unless added). A symmetric boxplot could hide two clusters.
Where does a whisker end when there is an outlier?
At the most extreme data value that is not an outlier. Data 3, 4, 4, 5, 6, 7, 8, 15 minutes: IQR = 3.5, upper fence 12.75, so 15 is an outlier and the whisker ends at 8.

1.9 Comparisons of the Distributions for One Quantitative Variable

What must a comparison of two distributions include?
Shape, centre, variability and unusual features, with a comparative word, values for both groups, units and context in each sentence, on the same scale.
What is a z-score?
The number of standard deviations a value is above (+) or below (−) the mean: z = (x − μ) / σ, or z = (x − x̄) / s from a sample. It has no units. Mean 50 kg, σ = 4 kg: 61 kg has z = 2.75.
How do you find a value from a z-score?
x = μ + zσ. With μ = 200 points and σ = 20 points, z = −1.5 gives x = 200 − 30 = 170 points.
Why use z-scores to compare values on different scales?
z-scores have no units, so they compare relative standing. 78 (mean 70, σ = 4) has z = 2.00; 88 (mean 75, σ = 10) has z = 1.30, so the first is relatively better.
What do parallel boxplots show and hide?
They show centre, variability, outliers and skewness or symmetry, but hide clusters, gaps and n. Box width shows spread, not sample size.
Does a higher z-score always mean a better result?
No. It means further above the mean. For times, a lower value (more negative z) can be better.

1.10 The Investigative Question Revisited and Data Collection

What should a full investigative question include?
The variables to collect, the analysis, and what can be concluded about which population. It either asks to estimate a parameter with plausible values or to test a claim in a stated direction.
How do an observational study and an experiment differ?
In an observational study nothing is imposed; values are recorded. In an experiment the researcher assigns treatments and then measures a response.
What is a census?
A study that records every individual in the population. Census versus sample is about who is measured; observational versus experiment is about how the data arise.
How do prospective and retrospective studies differ?
Both choose units now. Prospective studies record data now and into the future; retrospective studies gather data from the past.
What are factors, treatments and response variables?
A factor (explanatory variable) has levels imposed on experimental units. A treatment is one level or combination of levels; the number of treatments is the product of the levels (4 diets × 2 exercise plans = 8). The response is the outcome measured.
What is a confounding variable?
A variable associated with both the explanatory and the response variable, giving another explanation for the result. Show both links in context.
What does random selection or random assignment allow you to conclude?
Random selection allows generalising to the population sampled. Random assignment in an experiment allows a cause-and-effect conclusion. Volunteers allow generalising only to individuals like them.

1.11 Random Sampling

What is a simple random sample?
A sample chosen so that every possible sample of size n is equally likely. Label 1 to N and choose n different labels by a chance process; it is not defined by each individual having the same chance.
How is a stratified random sample taken?
Split the population into strata of similar individuals, take an SRS within every stratum and combine. Useful when groups differ on the variable. Proportional allocation: 300 boarders and 200 day students, n = 50, gives 30 and 20.
How is a cluster sample taken?
Split the population into clusters that each mirror the population, take an SRS of clusters, and measure everyone in the chosen clusters. Useful when the population is spread out and visits are costly.
How is a systematic random sample taken?
k ≈ N ÷ n, choose a random start from 1 to k, then every k-th individual. 500 names, n = 25: k = 20; start 13 gives 13, 33, 53, …, 493.
How do you use random digits to choose a sample?
Give all labels the same number of digits (001 to 250), skip numbers that are not labels, and skip repeats unless sampling with replacement.
Are stratified and systematic samples simple random samples?
No. They are random samples, but some samples of size n cannot happen, so they are not SRSs.
What is the difference between strata and clusters?
Strata are groups of similar individuals, and you sample within every stratum. Clusters each mirror the population, and you sample whole clusters, taking everyone in them.

1.12 Potential Problems with Sampling

What is bias in sampling?
A systematic error in the sampling method, so the statistic tends to be consistently above or consistently below the parameter. Chance variation misses by small amounts on both sides and is not bias.
What is undercoverage?
Part of the population is left out of, or less likely to be in, the sampling frame, as when an online-only survey misses people without internet.
How do voluntary response bias and nonresponse bias differ?
Voluntary response: people choose themselves, so strong opinions are over-represented. Nonresponse: the researcher chose people but some do not reply, and they differ from those who do.
What is response bias?
Answers or measurements drift one way from the truth: a leading question, self-reported behaviour, or a scale that reads heavy. "Wouldn't you agree homework is a waste of time?" likely makes the proportion too high.
Does a larger sample remove bias?
No. It only makes a biased estimate more precise. Random selection prevents bias in who is chosen, but nonresponse, response bias and an incomplete list can still bias a random sample.
How should you describe bias in an answer?
Name the kind of bias, the people missed or over-represented in context, and the likely direction (estimate too high or too low) and why.
What is a convenience sample?
A nonrandom sample of the easiest-to-reach individuals, such as asking only your own class. It is open to bias.

1.13 Experimental Design

What are the principles of a well-designed experiment?
Comparison of at least two treatments, random assignment, replication (more than one unit per treatment) and direct control of known extraneous variables.
What is a control group?
A group giving a baseline for comparison: no treatment, a placebo or the standard treatment. Direct control (fixing conditions) is different from a control group.
How do you calculate the placebo effect?
Mean response to placebo − mean response to no treatment. Placebo 4.5 and no treatment 3.0 gives 1.5 points. It is not treatment − placebo.
What are single-blind and double-blind experiments?
Single-blind: one side (participants or the researchers who meet them) does not know the treatments. Double-blind: neither side knows.
Why use random assignment in an experiment?
It spreads extraneous variables roughly evenly across groups, reducing confounding, and allows cause-and-effect conclusions. It is different from random sampling.
What are randomized block and matched pairs designs?
Block: group similar units into blocks, then randomise every treatment within each block, removing the blocking variable's variation. Matched pairs: two treatments with similar pairs, or each unit gets both in random order.
How should random assignment be described?
So someone could repeat it: label the units, say how labels are chosen and how repeats are handled, and which labels go to which treatment. "Randomly split them" is too vague.

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