Which way the tail points • How skew moves the mean • Compare mean, median, and mode
Data are skewed when one tail is longer than the other. Name the skew by the tail, not by the peak. How does the tail change the mean compared with the median and the mode?
Mean
The average. A long tail or an outlier pulls the mean toward it.
Median
The middle value after you line the data up. On a graph, half the area of the curve is on each side.
Mode
The value that appears most often. On a graph it is the peak — the highest point.
Memory hook
The mean follows the tail. The mode stays at the peak. The median sits between them.
Symmetric (bell-shaped)
Tails match. Peak in the center.
mean = median = mode
Right-skewed
Long tail toward high values (peak on the left).
mode < median < mean
Left-skewed
Long tail toward low values (peak on the right).
mean < median < mode
The graph shows a right-skewed distribution of test scores. Which statement is true?
The tail points right (toward high scores). The peak on the left is the mode. The mean is dragged into the tail, so it is the farthest right.
Order from smallest to largest: mode, then median, then mean.
Answer: A
The graph is approximately symmetric (bell-shaped). Which statement is true?
A symmetric (bell-shaped) graph has matching tails, so nothing pulls the mean left or right. The peak, the middle, and the average are all at the same center line.
Answer: B
The graph below is left-skewed. Which comparison of the mean, median, and mode is correct?
Left-skewed means the long tail points left (toward low scores). The peak on the right is the mode (largest). The mean is pulled into the low tail (smallest). The median is in between.
Answer: C
A teacher gives an easy quiz. Most students score near 90, and a few very low scores create a long tail to the left. Which statement must be true?
Most scores pile up on the high end, so this is left-skewed. A few low scores pull the mean toward the tail. That makes the mean the smallest of the three centers: mean < median < mode.
So the mean is less than the median. A is the opposite of what skew does. C would need a symmetric bell curve.
Answer: B
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