Descriptive Statistics Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page to practice descriptive statistics skills that appear everywhere in math and data literacy: finding the mean, median, and mode, calculating the range, identifying quartiles \((Q_1, Q_3)\) and the interquartile range (IQR), building a five-number summary, reading a box-and-whisker plot, and interpreting frequency, relative frequency, and percent. The lesson also introduces outliers using the 1.5×IQR rule and the meaning of variance and standard deviation. If you want a refresher, click Start lesson to open a step-by-step guide with examples and quick checks.
How this descriptive statistics practice works
1. Take the quiz: answer the descriptive statistics questions at the top of the page.
2. Open the lesson (optional): review formulas, step-by-step methods, and common mistakes for mean, median, mode, quartiles, and IQR.
3. Retry: return to the quiz and apply the descriptive statistics steps immediately.
What you will learn in the descriptive statistics lesson
Data basics & vocabulary
How to order a data set and count values correctly
Frequency and relative frequency for interpreting lists and tables
Core language: quartiles, percent, five-number summary, and outliers
Measures of center
Compute and interpret mean, median, and mode
Choose a good "typical value" when data has outliers or is skewed
Common errors: forgetting to sort before finding the median
Measures of spread
Find the range (max - min) for overall spread
Find quartiles and the interquartile range (IQR) for robust spread
Connect IQR to box plots and outlier detection
Box plots, outliers & standard deviation
Build a five-number summary and read a box-and-whisker plot
Identify outliers with the 1.5×IQR rule
Understand variance and standard deviation as measures of variability
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing descriptive statistics.
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Descriptive Statistics
Step-by-step guide
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Descriptive Statistics Lesson
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Lesson Overview
Lesson overview
Purpose: Build a clear understanding of descriptive statistics so you can summarize a data set, compare groups, and interpret “typical value” and “spread” using reliable methods.
Success criteria
Organize a data set by sorting values and counting observations correctly.
Calculate and interpret mean, median, and mode.
Calculate range as \( \text{max} - \text{min} \).
Find quartiles \(Q_1\) and \(Q_3\), then compute interquartile range (IQR) as \(Q_3 - Q_1\).
Create a five-number summary (min, \(Q_1\), median, \(Q_3\), max) and connect it to a box-and-whisker plot.
Use relative frequency and percent to describe how common a value or category is.
Identify outliers with the 1.5×IQR rule.
Understand variance and standard deviation as measures of variability.
Key vocabulary
Data set: a list of observations (numbers) you want to summarize.
Mean: the average, \( \bar{x}=\dfrac{\sum x}{n} \).
Median: the middle value when the data is ordered (or the average of the two middle values).
IQR: \(Q_3-Q_1\), the spread of the middle 50% of the data.
Outlier: a value far from the rest of the data (often checked with 1.5×IQR fences).
Quick pre-check
Pre-check 1: What is the median of \(\{7,8,9\}\)?
Hint: The median is the middle value after ordering the data.
Pre-check 2: Given the data set \(\{5,7,9\}\), what is the mean?
Hint: Mean = (sum of values) ÷ (number of values).
Measures of Center
Mean, median, and mode
Learning goal: Calculate mean, median, and mode, and know what each one tells you about “typical value.”
Key idea
The mean (average) uses every value: \[ \bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}. \] The median is the middle value after sorting the data (or the average of the two middle values when \(n\) is even). The mode is the most frequent value (data can be bimodal or have no mode). In skewed data or with outliers, the median is usually more resistant than the mean.
Worked example
Example: Find the mean, median, and mode of \(\{2,4,4,9,11\}\).
The data is already sorted. Mean: \[ \bar{x}=\frac{2+4+4+9+11}{5}=\frac{30}{5}=6. \] Median (middle of 5 values) is \(4\). Mode (most frequent) is \(4\).
Try it
Try it 1: What is the median of the data set \(\{1,2,3,4\}\)?
Hint: With 4 values, the median is the average of the two middle values.
Try it 2: What is the mean of \(\{5,10,15\}\)?
Hint: Add the values, then divide by 3.
Summary
Mean uses every value, but can be pulled by outliers.
Median is the middle of ordered data and is resistant to outliers.
Mode describes the most frequent value(s), especially useful for categories.
Measures of Spread
Range, quartiles, and interquartile range (IQR)
Learning goal: Measure variability using range and IQR, and compute \(Q_1\) and \(Q_3\) correctly.
Key idea
The range is the simplest measure of spread: \[ \text{Range}=\text{max}-\text{min}. \] Quartiles divide ordered data into four parts. The interquartile range is: \[ \text{IQR}=Q_3-Q_1, \] which measures the spread of the middle 50% of the data (and is less affected by outliers than the range).
Worked example
Example: For \(\{10,20,30,40\}\), find range, \(Q_1\), \(Q_3\), and IQR.
Range \(=40-10=30\). Median \(Q_2\) is the average of 20 and 30: \(25\). Lower half \(\{10,20\}\) gives \(Q_1=\frac{10+20}{2}=15\). Upper half \(\{30,40\}\) gives \(Q_3=\frac{30+40}{2}=35\). So \(\text{IQR}=35-15=20\).
Try it
Try it 1: What is the range of \(\{2,5,9\}\)?
Hint: Range = max − min.
Try it 2: What is the interquartile range of \(\{1,2,3,4,5,6,7,8,9\}\)?
Hint: Find \(Q_1\) and \(Q_3\) from the ordered list, then compute \(Q_3-Q_1\).
Summary
Range measures overall spread, but is sensitive to extreme values.
IQR measures spread of the middle 50% and is more resistant to outliers.
Frequency & Percent
Frequency, relative frequency, and percent
Learning goal: Convert between counts, fractions, decimals, and percent to describe how common something is.
Key idea
Frequency is a count. Relative frequency is the fraction of the total: \[ \text{Relative frequency}=\frac{\text{count}}{n}. \] To convert to a percent, multiply by 100: \[ \text{Percent}= \left(\frac{\text{count}}{n}\right)\cdot 100\%. \] This helps you compare groups even when sample sizes are different.
Worked example
Example: In \(\{0,1,0,1,1\}\), what fraction and percent are ones?
There are 3 ones out of 5 values, so the fraction is \(\frac{3}{5}\). As a decimal, \(\frac{3}{5}=0.6\), so the percent is \(0.6\cdot 100\% = 60\%\).
Try it
Try it 1: In the list \(\{0,1,0,1,1\}\), what fraction are ones?
Hint: Count the ones, then divide by the total number of values.
Try it 2: What percent of 50 is 15?
Hint: Compute \(\frac{15}{50}\) and convert to a percent.
Summary
Relative frequency is “part of the whole”: \(\frac{\text{count}}{n}\).
Percent is relative frequency \(\times 100\%\).
Five-Number Summary
Five-number summary and box-and-whisker plots
Learning goal: Build a five-number summary and connect it to a box plot to visualize center and spread.
Key idea
A five-number summary describes a data set using: \[ \text{min},\ Q_1,\ \text{median }(Q_2),\ Q_3,\ \text{max}. \] A box-and-whisker plot uses these five numbers: the box runs from \(Q_1\) to \(Q_3\), the median is a line inside the box, and the whiskers extend toward the minimum and maximum (or toward the non-outlier values, depending on the convention).
Worked example
Example: Find the five-number summary of \(\{2,4,6,8\}\).
Sorted: \(\{2,4,6,8\}\). Min \(=2\), max \(=8\). Median \(=\frac{4+6}{2}=5\). Lower half \(\{2,4\}\) gives \(Q_1=\frac{2+4}{2}=3\). Upper half \(\{6,8\}\) gives \(Q_3=\frac{6+8}{2}=7\). Five-number summary: \(2,\ 3,\ 5,\ 7,\ 8\).
Try it
Try it 1: What is the third quartile \(Q_3\) of \(\{10,20,30,40\}\)?
Hint: \(Q_3\) is the median of the upper half of the ordered data.
Try it 2: What is the first quartile \(Q_1\) of \(\{1,2,3,4,5,6\}\)?
Hint: With 6 values, the lower half is \(\{1,2,3\}\). Its median is \(Q_1\).
A box plot visualizes the middle 50% (the box) and the overall spread (the whiskers).
Outliers & Midrange
Outliers, the 1.5×IQR rule, and midrange
Learning goal: Identify outliers using IQR fences and understand the midrange as a “midpoint of extremes.”
Key idea
A common outlier check uses IQR fences: \[ \text{Lower fence}=Q_1-1.5(\text{IQR}), \quad \text{Upper fence}=Q_3+1.5(\text{IQR}). \] Values outside the fences are often treated as outliers. Another measure you may see is the midrange, the midpoint of the minimum and maximum: \[ \text{Midrange}=\frac{\text{min}+\text{max}}{2}. \] Midrange is easy to compute, but it depends only on the extremes (so it is not resistant to outliers).
Worked example
Example: Is \(30\) an outlier in \(\{1,2,3,4,5,6,7,8,30\}\) using the 1.5×IQR rule?
The median is \(5\). Lower half \(\{1,2,3,4\}\) gives \(Q_1=\frac{2+3}{2}=2.5\). Upper half \(\{6,7,8,30\}\) gives \(Q_3=\frac{7+8}{2}=7.5\). IQR \(=7.5-2.5=5\). Fences: lower \(=2.5-1.5(5)=2.5-7.5=-5\), upper \(=7.5+7.5=15\). Since \(30>15\), the value \(30\) is an outlier by the 1.5×IQR rule.
Try it
Try it 1: Which measure is the midpoint of the data range?
Hint: It uses only the minimum and maximum: \(\frac{\text{min}+\text{max}}{2}\).
Try it 2: If \(Q_1=10\) and \(Q_3=20\), which value is an outlier using the 1.5×IQR rule?
Hint: IQR \(=20-10=10\). Upper fence \(=20+1.5(10)=35\). Outliers are above 35 or below \(-5\).
Summary
Outlier fences: \(Q_1-1.5(\text{IQR})\) and \(Q_3+1.5(\text{IQR})\).
Midrange is easy but depends only on the extremes.
Standard Deviation
Variance, standard deviation, and z-scores
Learning goal: Understand standard deviation as a measure of typical distance from the mean and compute simple examples correctly.
Key idea
Variance measures average squared distance from the mean, and standard deviation is its square root. For a population (using all values), one definition is: \[ \sigma^2=\frac{\sum (x-\mu)^2}{n}, \quad \sigma=\sqrt{\sigma^2}. \] A z-score measures how many standard deviations a value is from the mean: \[ z=\frac{x-\mu}{\sigma}. \]
Worked example
Example: For the data set \(\{0,2\}\), find the population standard deviation and the z-score of \(x=2\).
Mean \(\mu=\frac{0+2}{2}=1\). Deviations: \(0-1=-1\), \(2-1=1\). Squares: \(1\) and \(1\). Variance \(\sigma^2=\frac{1+1}{2}=1\). Standard deviation \(\sigma=\sqrt{1}=1\). Z-score for \(x=2\): \(z=\frac{2-1}{1}=1\).
Try it
Try it 1: What is the population standard deviation of \(\{0,2\}\)?
Hint: Compute the mean, square the deviations, average them, then take the square root.
Try it 2: If the mean is \(10\) and the standard deviation is \(2\), what is the z-score of \(x=14\)?
Hint: Use \(z=\frac{x-\mu}{\sigma}\).
Summary
Standard deviation describes a typical distance from the mean.
Z-scores compare values across different scales by measuring “how many standard deviations away.”
Applications & Choices
Choosing the right statistic and interpreting data
Learning goal: Choose appropriate descriptive statistics and finish with a final check.
Where descriptive statistics show up
School and testing: summarize scores with mean/median and compare spread with IQR.
Science and experiments: report “typical result” and variability using standard deviation.
Sports and performance: compare consistency (low spread) vs volatility (high spread).
Data literacy: interpret charts like box plots and frequency tables in reports.
Worked example: median from unsorted data
Example: Find the median of \(\{5,2,9,4,7\}\).
First sort the data: \(\{2,4,5,7,9\}\). The middle (third) value is \(5\), so the median is \(5\).
Try it
Try it 1: What is the median of \(\{5,2,9,4,7\}\)?
Hint: Sort first, then choose the middle value.
Try it 2: What is the mode of \(\{2,2,3,4,4\}\)?
Hint: The mode is the value (or values) that occur most often. Here, two values tie.
Final recap
Center: mean, median, mode describe typical value in different ways.
Spread: range and IQR describe variability; IQR is more resistant to outliers.
Five-number summary and box plots summarize data visually and support outlier checks.
Standard deviation measures typical distance from the mean; z-scores compare across scales.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the descriptive statistics skill you need.
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