Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Skewness asks how a value or feature behaves inside the full distribution.
Common stuck point:Students often know a procedure related to skewness but skip the recognition step: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary? That leads to a calculation or graph that looks reasonable but answers a different question.
Sense of Study hint:Ask: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?
Worked Examples
Example 1
medium
A dataset has mean xΛ=12, median =10, and mode =8. Describe the skew and justify using the order of these three measures.Mean = 12, Median = 10, Mode = 8. Describe the skew.
Answer
rightΒ (positive)Β skew
First step
1
Order: mode < median < mean (8<10<12).
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Example 2
medium
A dataset has mean 25, median 30, mode 35. Describe the skew using the order of mean, median, and mode.Mean = 25, Median = 30, Mode = 35. Describe the skew.
Example 3
medium
Data: {1,1,2,2,3,3,4,4,5,5}. Compute the skew direction (no need to plug into the formula).Data: {1,1,2,2,3,3,4,4,5,5}. What is the skew direction?
Example 4
medium
A sample skewness is reported as β2.4. Briefly describe the shape.Sample skewness = β2.4. Describe the shape.
Example 5
hard
Compute the sample skewness of the data {1,2,3,4,10}. Use skew=(nβ1)(nβ2)nββ(sxiββxΛβ)3 and round to two decimals.Data: {1, 2, 3, 4, 10}. Compute the sample skewness.
Example 6
hard
A dataset of 100 salaries has skewness +3. A single $5M salary is removed. Most likely the new skewness is closer to which: +3, +1.5, or 0?
Example 7
hard
A QQ-plot against a normal distribution bows upward at the right end. What does this say about the data's skew?
Example 8
challenge
An exponential distribution with rate Ξ» has theoretical skewness 2. Why does this number not depend on Ξ»?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
A distribution with a long tail to the RIGHT has what type of skew?Which direction does the long tail point?
Example 2
easy
A distribution with a long tail to the LEFT has what type of skew?Which direction does the long tail point?
Example 3
easy
What is the skewness of a perfectly symmetric distribution?
Example 4
easy
For a right-skewed distribution, is the mean greater or less than the median?Right-skewed: mean > median. How do they compare?
Example 5
easy
For a left-skewed distribution, is the mean greater or less than the median?Left-skewed: mean < median. How do they compare?
Example 6
easy
Income data with a few billionaires typically has which skew?
Example 7
easy
Does the sign of the skewness tell you the direction of the longer tail?
Example 8
easy
Are outliers likely to have a large effect on skewness?
Example 9
medium
A data set has mean 60 and median 50. Describe its skew.Mean = 60, Median = 50. What type of skew?
Example 10
medium
A data set has mean 40 and median 48. Describe its skew.Mean = 40, Median = 48. What type of skew?
Example 11
medium
For a right-skewed data set, which is a better measure of center: mean or median? Why?
Example 12
medium
A symmetric data set's values are all multiplied by β1. What is its new skewness?
Example 13
medium
A right-skewed data set has all values negated (xβ¦βx). What happens to its skewness?
Example 14
medium
Exam scores cluster near the top with a few low outliers. What skew, and how do mean and median compare?Scores cluster near the top with a few low outliers. What skew?
Example 15
medium
Why does the skewness formula cube the standardized deviations rather than square them?
Example 16
medium
A distribution has skewness β1.2. Where does its longer tail point, and how do mean and median compare?Skewness = β1.2. Where does the longer tail point?
Example 17
medium
A distribution has skewness +0.9. Where does the longer tail point, and how do mean and median compare?Skewness = +0.9. Where does the longer tail point?
Example 18
challenge
For data 2,3,3,4,4,4,5,5,100, predict the skew direction and the order of mean, median, mode.
Example 19
challenge
Two data sets have the same mean and median. Set A's mean equals its median; set B's mean exceeds its median. Which is more skewed, and in which direction?
Example 20
challenge
Explain why the median is unaffected by adding one extreme high value while skewness and the mean both increase.
Example 21
easy
A histogram has a long thin tail stretching toward larger values on the right. What is its skew?What type of skew does this histogram show?
Example 22
easy
The skewness of a perfectly normal distribution equals what value?
Example 23
easy
Test scores where most students got near the maximum and only a few got very low scores would be skewed which way?Most students scored near the maximum; few scored very low. Which skew?
Example 24
easy
House prices in a city: most homes \$200k-\$500k, a few mansions worth \$5M. Which skew?Most homes $200kβ$500k; a few mansions at $5M. Which skew?
Example 25
easy
Adult shoe sizes from a mixed-gender population (with a small cluster of very large sizes) tend to skew which way?Adult shoe sizes from a mixed-gender population with some very large sizes. Which skew?
Example 26
medium
For each value xiβ in a data set, you compute (sxiββxΛβ)3 and the sum is negative. What does the sign tell you about the skew?
Example 27
medium
Why is the mean a poor center for a strongly right-skewed dataset?
Example 28
medium
A right-skewed dataset is transformed by taking log(xiβ) of every value. What typically happens to the skewness?
Example 29
medium
Every value in a left-skewed dataset is increased by 10. How does the skewness change?
Example 30
medium
Every value in a right-skewed dataset is multiplied by 3 (all values positive). How does the skewness change?
Example 31
medium
A right-skewed dataset has every value negated (xiβββxiβ). What kind of skew does the new dataset have?
Example 32
medium
Reaction times in a psychology experiment usually skew which way, and why?Reaction times in a psychology experiment. Which way do they skew?
Example 33
hard
For the data {2,4,6,8,10}, what is the sample skewness?Data: {2, 4, 6, 8, 10}. What is the sample skewness?
Example 34
hard
Two datasets have the same mean and standard deviation, but one has skewness +1.5 and the other β1.5. Are they identical in shape?
Example 35
hard
A teacher claims, 'My class scored higher than the median, so I'm above average.' If scores are strongly right-skewed, why is this reasoning suspect?
Example 36
hard
A dataset is bimodal, with two equal-height peaks at 20 and 80. What can you say about its skewness?
Example 37
hard
Pearson's second skewness coefficient is Sk2β=s3(xΛβmedian)β. A class has xΛ=72, median =78, s=6. Compute Sk2β.Class: mean = 72, median = 78, s = 6. Compute Pearson's Skβ.
Example 38
challenge
A researcher reports a sample skewness of +0.05 for n=20 observations and concludes the population is right-skewed. Why is this conclusion premature?