Language of Statistics
Fill in the blanks: Statistical thinking means recognizing that ________ is everywhere and that conclusions drawn from data always involve some ________.
Variation; uncertainty
The first 50 people who walk into a sporting event are asked about their TV preferences. What sampling method is this?
Convenience sampling.
Which type of graph should you use to show the favorite streaming services of 200 college students?
A bar graph, because the data are categorical.
For qualitative data such as favorite color, which measure of center is the only one that makes sense?
The mode.
A child's height is at the 90th percentile. What does that mean?
The child is taller than about 90% of children the same age.
What does a standard deviation of 0 tell you about a data set?
Every data value is the same (there is no spread).
An instructor wants the mean number of days her college's math students are absent in a quarter. Her sample has a mean of 3.5 days. Is 3.5 a parameter or a statistic?
A statistic. It describes a sample. A parameter describes the whole population and is usually unknown.
A researcher divides a university's students by major, then randomly selects 20 students from each major. What sampling method is this?
Stratified sampling.
To draw a simple random sample in Excel, we filled a helper column with random decimals, sorted by that column, and took the first n rows. Which Excel function made the random decimals?
RAND
At a company, the mean salary is $95,000 and the median salary is $52,000. What shape is the salary distribution, and why?
Right-skewed. A few very high salaries pull the mean above the median.
A runner's 5K finish time is at the 90th percentile. Is that good or bad? Explain.
Bad. A lower time is better, and her time was slower than about 90% of the runners.
Of the range, IQR, variance, and standard deviation, which measure of spread is resistant to outliers?
Only the IQR.
A pollster uses a sample of 200 voters to predict the winner of a statewide election. Is this descriptive or inferential statistics?
Inferential. It uses a sample to draw a conclusion about a larger population.
A city is divided into 10 neighborhoods. Three neighborhoods are chosen at random, and every household in those three is surveyed. What sampling method is this?
Cluster sampling.
What are the key differences between a histogram and a bar graph?
A histogram shows quantitative data grouped into intervals, with bars that touch and must stay in numerical order. A bar graph shows categorical data, with gaps between the bars, and the bars can be put in any order.
Using the procedure in the book, find the median of: 12, 15, 17, 22, 28, 33.
19.5. There is an even number of values, so average the two middle ones: (17 + 22)/2.
A data set has Q1 = 10 and Q3 = 20. Find the outlier fences using the IQR rule. Is a value of 38 an outlier?
IQR = 10. Lower fence = 10 − 1.5(10) = −5. Upper fence = 20 + 1.5(10) = 35. Yes, 38 is an outlier because it is above 35.
Find the sample variance and the sample standard deviation of 6, 10, 14.
Mean = 10. Squared deviations: 16, 0, 16, which sum to 32. s² = 32/(3 − 1) = 16, so s = 4.
Give the level of measurement (nominal, ordinal, interval, or ratio) for each: (a) letter grades A, B, C, D, F (b) baking temperatures in °F (c) income in dollars.
(a) Ordinal (b) Interval, because 0°F is not a true zero (c) Ratio.
What is a double-blind experiment, and what does the blinding protect against?
Neither the participants nor the researchers who measure outcomes know who got which treatment. It guards against the placebo effect and against researcher bias in measuring results.
Name the Excel function we used in class for each task: (a) finding the size of each stratum (b) randomly picking a cluster.
(a) COUNTIF (b) RANDBETWEEN
Estimate the mean from this grouped frequency table: 0–4 : 2;
5–9 : 6;
10–14 : 4.
Midpoints 2, 7, 12. Mean ≈ (2·2 + 6·7 + 4·12)/12 = 94/12 ≈ 7.83.
Using the procedure in the book, find P35 for this data: 34, 12, 45, 26, 57, 21, 38, 15, 48, 30, 23, 40, 18, 52, 31, 43, 36.
28. Sorted: 12 15 18 21 23 26 30 31 34 36 38 40 43 45 48 52 57 (n = 17). i = 0.35(18) = 6.3, which is not a whole number, so average the 6th and 7th values: (26 + 30)/2 = 28.
Student A scored 90 in a class with mean 80 and SD 5. Student B scored 90 in a class with mean 75 and SD 10. Relative to their own class, whose score is more impressive?
Student A. z_A = (90 − 80)/5 = 2 and z_B = (90 − 75)/10 = 1.5, so A is farther above their class mean.
A cardiologist wants to know the mean recovery time of their heart attack patients. They randomly select 25 of these patients from their records and measure each patient's recovery time. Identify the population, sample, parameter, statistic, variable, and data.
Population: all of the cardiologist's heart attack patients.
Sample: the 25 selected patients.
Parameter: the mean recovery time of all of those patients.
Statistic: the mean recovery time of the 25 sampled patients. Variable: a patient's recovery time.
Data: the recovery times actually recorded (e.g., 42 days).
A study finds that cities with more fire stations have more fires. Can we conclude that fire stations cause fires? Explain, and say what kind of study would be needed to show cause and effect.
No. City size is a lurking (confounding) variable: bigger cities have more fire stations and more fires. This is an observational study. Only a randomized experiment supports a cause-and-effect conclusion.
A news channel shows last year's revenue ($10M) next to this year's ($11M), with the y-axis starting at $9.5M. This year's bar looks three times as tall. Why is this misleading, and what is the actual percent increase?
The y-axis is truncated, which exaggerates the difference between the bars. The actual increase is only 10% ($1M out of $10M).
A charity has 100 donors. One gave $1,000,000 and the other 99 each gave $100. Find the mean and the median donation. Which one better describes a typical donation, and why?
Mean = 1,009,900/100 = $10,099. Median = $100. The median is better because it is resistant: the one extreme donation doesn't pull it up.
Jesse is ranked 37th in a graduating class of 180 (rank 1 = highest GPA). Assuming no ties, use the procedure in the book to find his percentile rank.
80th percentile. 143 students are below him and 1 is equal (Jesse himself): (143 + 0.5·1)/180 × 100 ≈ 79.7, which rounds to 80.
IQ scores have mean 100 and SD 15. If the distribution is bell-shaped, about what percent of scores fall between 70 and 130?
70 to 130 is the mean ± 2 SD. About 95% (Empirical Rule).