Comparing and Combining Data: Quick Revision

A brief note on Comparing and Combining Data. Read it before a test. Try each question at the end before you open its answer.

Comparing

• Coefficient of variation: $\text{CV} = \dfrac{\sigma}{\bar{x}} \times 100\%$. The smaller CV is the more consistent data.
• Where to use it: data with a true zero, such as mass, length and time. Not for degrees Celsius.
• Change every value, $y = a x + b$: $\bar{y} = a\bar{x} + b$, $\sigma_y = \lvert a \rvert\,\sigma_x$, $\sigma_y^2 = a^2\sigma_x^2$.
• Adding a number: moves the mean and the median, and leaves every spread unchanged.

Combining

• Combined mean: $\bar{x} = \dfrac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$.
• Combined variance: $\sigma^2 = \dfrac{n_1\left(\sigma_1^2 + d_1^2\right) + n_2\left(\sigma_2^2 + d_2^2\right)}{n_1 + n_2}$, with $d_i = \bar{x}_i - \bar{x}$.
• Never average the standard deviations.
• A wrong reading: change $\sum x$ and $\sum x^2$, then find the mean and the variance again.

Quartiles and outliers

• Quartiles: $Q_1$ and $Q_3$ are the medians of the lower and the upper half. Leave the median out when $n$ is odd.
• Interquartile range: $\text{IQR} = Q_3 - Q_1$. It ignores extreme values.
• Five-number summary: minimum, $Q_1$, median, $Q_3$, maximum. The box plot draws it.
• Fences: $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$. A value outside them is an outlier.

A worked example

Class A has $30$ students with mean $60$ and standard deviation $8$. Class B has $20$ students with mean $70$ and standard deviation $6$.

1. The coefficients. $\text{CV}_A = \dfrac{8}{60} \times 100 = 13.33\%$ and $\text{CV}_B = \dfrac{6}{70} \times 100 = 8.57\%$. Class B is more consistent.
2. Both classes together, the mean. $\dfrac{30 \times 60 + 20 \times 70}{50} = 64$.
3. The variance. $d_1 = -4$ and $d_2 = 6$. So $\sigma^2 = \dfrac{30\,(64 + 16) + 20\,(36 + 36)}{50} = 76.8$.
4. A change. If every mark of class A is doubled and $10$ is added, the mean is $130$ and the standard deviation is $16$.

Practice with new numbers

A program to try

The program prints the coefficient of variation of each class, the effect of a change, and the combined mean and variance of the worked example.

Mistakes to avoid

• Averaging the standard deviations of two groups.
• Adding a shift to the standard deviation.
• Forgetting the absolute value when the multiplier is negative.
• Keeping the median in both halves for an odd number of values.
• Forgetting that a wrong reading changes $\sum x^2$.

Check yourself

1. Find the coefficient of variation when the mean is $50$ and the standard deviation is $6$.

2. The mean is $14$ and the standard deviation is $5$. Every value becomes $3x - 2$. Find the new mean and standard deviation.

3. Ten values have mean $8$ and thirty values have mean $12$. Find the mean of all forty.

4. The quartiles of a data set are $12$ and $20$. Find the upper fence.

5. Which of the mean, the median, the range and the standard deviation do not change when the same number is added to every value?

Sources

• OpenStax, Introductory Statistics, Chapter 2, sections on the spread of the data, box plots and outliers.
• NCERT, Mathematics for Class XI, Chapter 13 (Statistics).