Mean, median and mode together
For many tables, 3 Median is close to Mode + 2 Mean.
Three averages of one table
For the 30 students the book found a mean of 62 and a mode of 52. A third average, the median, comes from the cumulative frequencies.
The three numbers measure the middle of the same data in three different ways. Is there a link between them?
Do you think the three averages are always equal?
What this lesson covers
The idea
The three measures of central tendency are related empirically by 3 Median = Mode + 2 Mean.
Three averages of one table
For the 30 students the book found a mean of 62 and a mode of 52. A third average, the median, comes from the cumulative frequencies.
The three numbers measure the middle of the same data in three different ways. Is there a link between them?
Do you think the three averages are always equal?
- Yes, always equal
- No, but they are close to each other
- No, they can be very far apart
Place all three
For each table, tap Mean, Median and Mode. Each one is worked out with the formula you know and placed on the histogram. Then look at the two sides of the test 3 × Median and Mode + 2 × Mean.
An empirical relationship
3 Median = Mode + 2 Mean. This is an empirical relationship between the three measures of central tendency.
"Empirical" means found by looking at many data sets, not proved. It holds roughly. In your three tables the two sides were close but not equal. So the relation can give a rough value of one measure when the other two are known.
Notes
3 Median = Mode + 2 Mean is an empirical relationship: it holds roughly, not exactly.
Check yourself
The median of a data set is 62.5 and its mean is 62. Use 3 Median = Mode + 2 Mean to estimate the mode.
Answer: 63.5
Mode = 3 × 62.5 − 2 × 62 = 187.5 − 124 = 63.5. The real mode of this table is 52, so the relation is only a rough guide.
A data set has median 12 and mode 6. Use the relationship to find the mean.
Answer: 15
2 Mean = 3 × 12 − 6 = 30, so Mean = 15.
A data set has mean 58 and mode 52. Use the relationship to find the median.
Answer: 56
3 Median = 52 + 2 × 58 = 168, so Median = 56.
For the 30 students, 3 × Median = 187.5 and Mode + 2 × Mean = 176. What does this show?
- The relationship is empirical: close, but not exact — correct. Yes! The two sides are near each other but not equal. An empirical relationship holds only roughly.
- The mean was calculated wrongly. The mean 62 is right. The relationship itself is only approximate.
- The relationship is exact and a mistake was made. No. 3 Median = Mode + 2 Mean is empirical, so small differences are normal.