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D. Statistical Analysis · One-Variable Data Analysis

One-Variable Data Analysis: Unit Review

D1.1 D1.2 D1.3 D1.4 D1.5About 40 minutes
Learning goals
  • I can organize, graph and describe one-variable data, and choose summaries that suit its shape.
  • I can calculate and interpret measures of centre, spread and position, including standard deviation, percentiles and z-scores.
  • I can use the normal model, interpret margins of error, and recognize misleading summaries.

Words to know

standard deviation
A measure of spread: a typical distance from the mean (σ for a population, s for a sample).
z-score
The number of standard deviations a value is from the mean.
margin of error
The likely maximum difference between a sample statistic and the parameter, at a stated confidence level.

Let's review!

In this unit you organized one-variable data into frequency tables, chose and drew graphs to suit the type of data, described shapes, calculated measures of centre and spread (including the standard deviation for populations and samples), located values with quartiles, percentiles and z-scores, flagged outliers with box plots, modelled data with the normal distribution, explored sampling variability and margins of error, and practised interpreting, comparing and critiquing summaries.

This review has a few questions from every lesson. Try each one, then press Show answer to check. Next, the unit Two-Variable Data Analysis and Validity looks at relationships between two variables and at how statistics are used in the media.

Organizefrequency tables (1)Graphbar, circle, histogram (2, 3)Shapeskew, modes (4)Centremean, median, mode (5)Spreadσ, s, IQR (6, 7)Positionpercentiles, z (8, 10)Box plotsoutliers (9)Normal modelsamples, ME (11-13)Interpretcompare, mislead (14, 15)
The unit at a glance, with lesson numbers. Each step builds on the one before: organize, graph and describe, measure, locate, model, then interpret.

How to use this review

Work through one or two sections at a time with a scientific calculator (or a spreadsheet for checking). Sketch a quick graph whenever a question involves shape, a box plot or a normal curve; the sketch catches many errors. Keep full precision until the final answer. For every question you miss, write down why (wrong formula? population vs sample? percentile vs percent? forgot to check the shape?) and go back to that lesson. Finish with the 10-question quiz.

One-Variable Data and Frequency Distributions

1

Classify: (a) number of hockey games watched last month, (b) time to shovel the driveway, (c) favourite Canadian city.

2

Cumulative frequencies 6, 15, 27, 36, 40. Find the frequency of the third class and the percent of values in the first two classes.

3

Data ranges from 12 to 88 and you want about 8 classes. Suggest a width and the first class.

Graphing Categorical Data: Bar Graphs and Circle Graphs

1

In a survey of 80 people, 28 chose winter as their favourite season. Find the sector angle.

2

Why is a circle graph wrong for "Which winter sports do you play? (choose all that apply)"?

3

Club X: 9 of 30 members are in Grade 12. Club Y: 14 of 56. Which has the larger proportion of Grade 12s?

Histograms, Frequency Polygons, Ogives and Stem-and-Leaf Plots

1

Classes [0, 5), [5, 10), [10, 15), [15, 20) have frequencies 3, 7, 6, 4. List the ogive points.

2

Use the ogive in the previous question to estimate the median.

3

In a frequency polygon, where are the points plotted?

The Shape of a Distribution: Symmetry, Skewness and Modes

1

Mean 34, median 27. Predict the shape.

2

Give a real-world variable that is likely skewed left, and why.

3

A histogram has two clear peaks. What should you do before summarizing?

Measures of Central Tendency: Mean, Median, Mode and Weighted Mean

1

Values 1, 2, 3, 4 have frequencies 2, 5, 8, 5. Find the mean.

2

Tests 40%, project 25%, exam 35%; marks 80, 90, 66. Find the final mark.

3

Class of 24 averaged 70, class of 16 averaged 80. Combined mean?

4

Median class [20, 30), n = 50, cumulative frequency before it 18, frequency 14. Estimate the median.

Measures of Spread: Range, IQR, Variance and Standard Deviation

1

Find σ for the population 2, 4, 6, 8.

2

Find s if 2, 4, 6, 8 is a sample.

3

When should you use s instead of σ?

Standard Deviation from Frequency Tables, Grouped Data and Technology

1

x = 1, 2, 3 with f = 5, 10, 5. Find μ and σ.

2

Mean 15, σ = 3. Each value becomes y = 2x − 4. New mean and σ?

3

A 1-Var Stats output shows Sx = 3.8 and σx = 3.6 for a sample of 12 students. Which do you report?

Quartiles, Percentiles and Percentile Rank

1

Data: 4, 7, 9, 10, 12, 15, 18, 21 (n = 8). Find Q1, median, Q3.

2

Same data: find P75 with the position rule.

3

In a group of 60, 41 scored below Kai and 3 tied with him. Find his percentile rank.

Box Plots, Outliers and Their Effects

1

Q1 = 30, Q3 = 42. Find the fences.

2

Using the review box plots below, compare Set P and Set Q.

3

Removing an extreme outlier changes which statistics the most?

Review: compare these two box plotsSet P1220242733Set Q822303437510152025303540value
Box plots for the box-plot review question. Read the five-number summaries from the labels.

z-Scores: Measuring Position in Standard Deviations

1

μ = 80, σ = 5. Find z for 72.

2

μ = 80, σ = 5. Find the value with z = 2.4.

3

Ana: 78 in a class with μ 70, σ 5. Ben: 84 in a class with μ 78, σ 3. Who did better relative to their class?

Modelling Data with the Normal Distribution

1

Normal, μ = 50, σ = 4. What percent is between 42 and 58 (by the 68–95–99.7 rule)?

2

Same model: what percent is above 55? (P(Z < 1.25) = 0.8944)

3

Same model: find P90 (z ≈ 1.28).

4

A data set has 55% of values within 1 s of the mean. Is a normal model suitable?

Samples, Statistics and Sampling Variability

1

Is "the mean of a random sample of 200 seedlings" a parameter or a statistic?

2

Find SE(p̂) for p = 0.3 and n = 84.

3

σ = 12. What sample size makes SE(x̄) = 2?

Margin of Error and Confidence Level

1

p̂ = 0.62, n = 600. Find the 95% margin of error.

2

What sample size gives ±4 points at 95% confidence with p unknown?

3

A poll: 47% vs 43%, ±3 points, 19 times out of 20. Can it say who leads?

Interpreting and Comparing One-Variable Data Sets

1

Which summaries should describe a skewed-right data set with an outlier?

2

Write one sentence comparing: Group A median 52 kg, IQR 6 kg; Group B median 58 kg, IQR 11 kg.

3

Name a limitation for a comparison based on two samples of 12 fish each.

How Statistical Summaries Can Mislead

1

A bar graph of 30 and 33 starts at 28. How many times taller does the 33 bar look?

2

A tax rate rises from 8% to 10%. Describe the change two ways.

3

A mean wait time of 20 min combines 500 patients averaging 15 min and 100 averaging 45 min. Check the mean and say what it hides.

Just for fun

Unit review joke: a statistician's ideal holiday is in Normal, Illinois, where the weather is never more than three standard deviations from pleasant.

Check your understanding

Quiz

1.

A sample of 5 values has Σ(x − x̄)² = 36. What is s?

2.

Which graph best displays the distribution of 500 students' reaction times?

3.

A distribution has mean 64 and median 71. It is most likely

4.

A value has z = 1.5 in a distribution with μ = 40 and σ = 6. What is the value?

5.

Q1 = 18 and Q3 = 26. Which value is an outlier?

6.

In a normal distribution, about what percent of values are more than 1σ below the mean?

7.

A pollster wants to halve her margin of error. She should

8.

In 40 values, 25 are below x and 2 equal x. The percentile rank of x is

9.

Every value of a data set is increased by 10. Which statistic stays the same?

10.

Which is the most honest summary of house prices in a city with a few very expensive mansions?

For parents and teachers

How to use the review results. If your teen misses a question, send them back to the lesson named in that section's title:

  • Frequency tables, relative or cumulative frequency, class widths → One-Variable Data and Frequency Distributions.
  • Choosing a graph, sector angles, percents for groups of different sizes → Graphing Categorical Data.
  • Histograms, polygons, ogives, stem-and-leaf plots → Histograms, Frequency Polygons, Ogives and Stem-and-Leaf Plots.
  • Naming skew, predicting mean vs median → The Shape of a Distribution.
  • Mean from a table, weighted means, grouped estimates → Measures of Central Tendency.
  • Variance and standard deviation, σ vs s → Measures of Spread; frequency tables, technology and transformations → Standard Deviation from Frequency Tables, Grouped Data and Technology.
  • Quartiles, percentiles, percentile rank → Quartiles, Percentiles and Percentile Rank.
  • Fences, outliers, reading box plots → Box Plots, Outliers and Their Effects.
  • z-scores → z-Scores; normal percents and percentiles → Modelling Data with the Normal Distribution.
  • Parameters vs statistics, standard error → Samples, Statistics and Sampling Variability; margins of error and sample size → Margin of Error and Confidence Level.
  • Weak written comparisons, wrong choice of summary → Interpreting and Comparing One-Variable Data Sets; graph tricks, percent vs points → How Statistical Summaries Can Mislead.

The most common errors across the unit are: using σ when the data is a sample, confusing percentiles with percents, forgetting to check the shape before using the mean or a normal model, and comparing raw counts for groups of different sizes.