Statistics

11, 14, 22, 19, 23, 22, 22 Find the mean, median, mode, range, variance, and standard deviation.

Step-by-step solution with explanation

Final Answer

Mean = 19 | Median = 22 | Mode = 22 | Range = 12 | Variance ≈ 18.86 | Standard Deviation ≈ 4.34

Step-by-step solution

1

Sort the data and find the count

Sorting the values from least to greatest makes it easier to find the median and range. We have 7 values total.
2

Calculate the mean (average)

\text{Mean} = \frac{11+14+22+19+23+22+22}{7} = \frac{133}{7} = 19$
Add all 7 values together to get 133, then divide by 7. The mean is 19.
3

Find the median (middle value)

With 7 values, the middle value is the 4th one. Counting in the sorted list, the 4th value is 22.
4

Find the mode (most frequent value)

22 appears 3 times, more than any other value. That makes it the mode.
5

Find the range (spread of data)

Subtract the smallest value from the largest value. This tells us how spread out the data is.
6

Calculate the variance

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Subtract the mean (19) from each value, square each result, add them all up, then divide by n = 7. This gives the population variance.
7

Calculate the standard deviation

Take the square root of the variance. The standard deviation tells us, on average, how far each value is from the mean.

Understanding this problem

Learning Insight

The mean, median, and mode are all ways to describe the 'center' of data, but they can give different answers. The variance and standard deviation measure how spread out the data is around that center — a small standard deviation means values cluster close to the mean, while a large one means they're spread far apart.

Quick Tip

Memory trick: Mean = Math (add then divide), Median = Middle, Mode = Most. For standard deviation, just remember it's always the square root of the variance.

Common Mistake

Students often forget to square the differences when computing variance, or they take the square root too early. Also, many students confuse population variance (divide by n) with sample variance (divide by n−1) — make sure you know which one your class uses.