← VCE Data Analytics U4O1

Skew: positive and negative

Data with a longer tail on one side. The tail pulls the mean its way: a right tail is positive skew, a left tail is negative skew, and the median stays nearer the bulk.

Part of: VCE Data Analytics - U4O1 - Development and Evaluation — Module 33

Understanding Skew: positive and negative in VCE Data Analytics U4O1

What it does: The formula tests whether the mean is greater than the median, which is the pattern for positive skew. The median is not pulled by the tail, so it is the better value to report as typical for skewed data. The skew is named after the direction of the longer tail. A tail to the right, towards high values, is positive skew. The long left tail of low values drags the mean down. The median stays closer to where most values sit.

Common mistakes: A long right tail pulls the mean above the median, so positive skew is when mean > median. The condition should be if mean > median:. The module is called statistics, with an s. Without it, statistics.mean() and statistics.median() fail. It should be import statistics.

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This module's key terms are also in a printable crossword (Modules 3–37) and word search (Modules 3–62).

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Learning

Duo

A data set has a long tail stretching to the right (high values). What is this called?

Positive (right) skewNegative (left) skew

Matches

The tail points left, so the data is positively skewed

↓

Skew is named after the direction of the tail. A left tail is negative skew, not positive skew.

Applying

Meanings

Term

skew

Meaning

Data with a longer tail on one side. The tail pulls the mean its way: a right tail is positive skew, a left tail is negative skew, and the median stays nearer the bulk.

Practice

=AVERAGE(D2:D8)-MEDIAN(D2:D8)

Testing

Gaps

=AVERAGE(A2:A40)>(A2:A40)

Check Answer

Spots

3median = 3

4if mean < median:

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Quick Questions

1. A data set has a long tail stretching to the right (high values). What is this called?

A. Negative (left) skew

B. Positive (right) skew ✓

The skew is named after the direction of the longer tail. A tail to the right, towards high values, is positive skew.

2. A data set has a long tail stretching to the left (low values). What is this called?

A. Positive (right) skew

B. Negative (left) skew ✓

A longer tail on the left, towards low values, is negative skew.

3. In positively skewed data, which statement is usually true?

A. The mean is greater than the median ✓

B. The mean is less than the median

The long right tail of high values pulls the mean up, towards the tail. The median stays nearer the bulk of the data.

4. In negatively skewed data, which statement is usually true?

A. The mean is less than the median ✓

B. The mean is greater than the median

The long left tail of low values drags the mean down. The median stays closer to where most values sit.

5. House prices in a suburb: most are $600k to $800k, but a few are over $5 million. Which skew is this?

A. Positive skew, because a few very high values form a right tail ✓

B. Negative skew, because the few very high values pull the data down

A few very high values create a long right tail. This is positive skew, and the mean will be higher than the median.

Match the Pairs

Mean > Median → A sign of positive (right) skew: a long tail of high values pulls the mean above the median

Mean < Median → A sign of negative (left) skew: a long tail of low values pulls the mean below the median

House prices have a few very expensive sales on the right, so the data is positively skewed and the mean is higher than the median → =AVERAGE(B2:B50)>MEDIAN(B2:B50)

For skewed data, the median is usually the better measure of the typical value because it stays nearer the bulk of the data → =MEDIAN(B2:B50)

The tail points left, so the data is positively skewed → Skew is named after the direction of the tail. A left tail is negative skew, not positive skew.

Data is skewed right, so the mean is below the median → In right (positive) skew the tail pulls the mean up, so the mean is above the median, not below it.

Key Term

skew

Data with a longer tail on one side. The tail pulls the mean its way: a right tail is positive skew, a left tail is negative skew, and the median stays nearer the bulk.

Fill in the Blanks

1. =AVERAGE(A2:A40)>MEDIAN(A2:A40)

Hint: Type the function that finds the middle value. It is compared with the mean to check for positive skew.

2. IF mean > median THEN OUTPUT "Positive skew" END IF

Hint: Type the comparison operator for when a long right tail pulls the mean above the median.

3. IF mean < median THEN OUTPUT "Negative skew" END IF

Hint: Type the comparison operator for when a long left tail pulls the mean below the median.

4. avg = statistics.mean(marks) mid = statistics.median(marks) print(avg > mid)

Hint: Type the statistics module function that finds the average, the measure the tail pulls.

5. IF skewed THEN centre ← median END IF

Hint: Type the measure of centre that stays nearer the bulk of skewed data, so it is better to report.

Spot the Error

Question 1

A data = [2, 3, 3, 4, 40]

B mean = sum(data) / len(data)

C median = 3

D if mean < median:

E print("positive skew")

Line D — A long right tail pulls the mean above the median, so positive skew is when mean > median. The condition should be if mean > median:.

Question 2

A import statistic

B data = [1, 50, 55, 58, 60]

C mean = statistics.mean(data)

D median = statistics.median(data)

E print(mean < median)

Line A — The module is called statistics, with an s. Without it, statistics.mean() and statistics.median() fail. It should be import statistics.

Question 3

A incomes = [30, 32, 35, 38, 150]

B mean = sum(incomes) / 5

C median = sorted(incomes)[4]

D print(mean, median)

Line C — With 5 sorted values the median is the middle one, at index 2. Index 4 is the largest value (the outlier), not the median. It should be [2].

Question 4

A data = [5, 6, 6, 7, 8, 30]

B tail = "left"

C skew = "positive"

D print(tail, skew)

Line B — The long tail stretches towards the large value 30, so the tail is on the right. A right tail means positive skew. It should be tail = "right".

Question 5

A mean = 72

B median = 80

C skew = "positive"

D print("Skew:", skew)

Line C — The mean (72) is below the median (80), so a left tail is pulling the mean down. That is negative skew. It should be skew = "negative".

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