Measures of Center and Variation in Statistics: Study Notes

Describing, Exploring, and Comparing Data

Measures of Center

Measures of center are statistical values that represent the central point or typical value of a data set. The most common measures are the mean, median, mode, and midrange.

• Mean (Arithmetic Mean): The sum of all data values divided by the number of values. It is the most widely used measure of center.

• Median: The middle value when the data set is ordered. If the number of values is even, the median is the mean of the two middle values.

• Mode: The value(s) that occur most frequently in the data set. A data set may have no mode, one mode (unimodal), two modes (bimodal), or more (multimodal).

• Midrange: The value halfway between the maximum and minimum values in the data set.

Properties of the Mean

• Sample means drawn from the same population tend to vary less than other measures of center.

• The mean uses every data value in its calculation.

• The mean is not resistant; a single extreme value (outlier) can substantially affect it.

Formulas:

• Sample mean: $x = \frac{\sum x}{n}$

• Population mean: $\mu = \frac{\sum x}{N}$

Where $\sum x$ is the sum of all data values, $n$ is the sample size, and $N$ is the population size.

Properties of the Median

• The median is resistant to extreme values; it does not change much with outliers.

• The median does not directly use every data value.

How to Find the Median:

1. Sort the data in order.

2. If the number of values is odd, the median is the middle value.

3. If even, the median is the mean of the two middle values.

Properties of the Mode

• The only measure of center that can be used with qualitative (categorical) data.

• A data set can have no mode, one mode, or multiple modes.

• Bimodal: Two values occur with the highest frequency.

• Multimodal: More than two values occur with the highest frequency.

Properties of the Midrange

• Very sensitive to extreme values (not resistant).

• Easy to compute.

• Rarely used in practice but useful for understanding different measures of center.

Formula:

• Midrange: $\text{Midrange} = \frac{\text{Maximum value} + \text{Minimum value}}{2}$

Rounding Rules for Measures of Center

• For mean, median, and midrange: round to one more decimal place than the original data values.

• For mode: do not round; report the value as it appears in the data.

• Round only the final answer, not intermediate calculations.

Mean from a Frequency Distribution

• Formula: $x = \frac{\sum (f \times x)}{\sum f}$

• Where $f$ is the frequency and $x$ is the class midpoint.

• This method gives an approximate mean for grouped data.

Weighted Mean

• Formula: $\bar{x} = \frac{\sum (w \times x)}{\sum w}$

• Where $w$ is the weight assigned to each value $x$.

Measures of Variation

Measures of variation describe how spread out the data values are. Common measures include the range, standard deviation, and variance.

Range

• Formula: $\text{Range} = \text{Maximum value} - \text{Minimum value}$

• Very sensitive to extreme values; does not reflect variation among all data values.

Standard Deviation

• The most commonly used measure of variation.

• Measures how much data values deviate from the mean.

• Always non-negative; zero only if all values are the same.

• Units are the same as the original data.

• Sample standard deviation ($s$) is a biased estimator of population standard deviation ($\sigma$).

Formulas:

• Sample standard deviation: $s = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}$

• Shortcut formula: $s = \sqrt{\frac{n \sum x^2 - (\sum x)^2}{n(n-1)}}$

• Population standard deviation: $\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$

Variance

• Variance is the square of the standard deviation.

• Sample variance: $s^2$

• Population variance: $\sigma^2$

• Units are the square of the original data units (e.g., ft2, sec2).

• Sample variance is an unbiased estimator of population variance.

Mean Absolute Deviation (MAD)

• Formula: $\text{MAD} = \frac{\sum |x - \bar{x}|}{n}$

• Represents the average distance of data values from the mean.

Range Rule of Thumb

• For many data sets, about 95% of values lie within 2 standard deviations of the mean.

• Significantly low values: $\mu - 2\sigma$ or lower.

• Significantly high values: $\mu + 2\sigma$ or higher.

• Estimate standard deviation: $s \approx \frac{\text{Range}}{4}$

Chebyshev's Theorem

• For any data set, the proportion of values within $k$ standard deviations of the mean is at least $1 - \frac{1}{k^2}$, for $k > 1$.

• For $k = 2$: At least 75% of values are within 2 standard deviations.

• For $k = 3$: At least 89% of values are within 3 standard deviations.

Coefficient of Variation (CV)

• Describes the standard deviation relative to the mean, expressed as a percentage.

• Sample: $\text{CV} = \frac{s}{\bar{x}} \times 100$

• Population: $\text{CV} = \frac{\sigma}{\mu} \times 100$

• Round CV to one decimal place (e.g., 25.3%).

z Scores (Standard Scores)

A z score indicates how many standard deviations a data value is above or below the mean.

• z scores have no units.

• A z score less than or equal to -2 is considered significantly low; greater than or equal to +2 is significantly high.

• If a value is less than the mean, its z score is negative.

Formulas:

• Sample: $z = \frac{x - \bar{x}}{s}$

• Population: $z = \frac{x - \mu}{\sigma}$

Round z scores to two decimal places (e.g., 2.31).

Percentiles, Quartiles, and the Five-Number Summary

Percentiles and quartiles are measures of location that divide data into groups with equal numbers of values.

• Percentiles (Pk): Divide data into 100 groups; P50 is the median.

• Quartiles (Q1, Q2, Q3): Divide data into four groups; Q1 is the 25th percentile, Q2 is the median (50th percentile), Q3 is the 75th percentile.

Five-Number Summary:

1. Minimum

2. First quartile (Q1)

3. Second quartile (Q2, median)

4. Third quartile (Q3)

5. Maximum

Other Measures Based on Quartiles and Percentiles

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

• Semi-interquartile Range: $\frac{Q_3 - Q_1}{2}$

• Midquartile: $\frac{Q_3 + Q_1}{2}$

• 10–90 Percentile Range: $P_{90} - P_{10}$

Odds and Payoff Odds

Odds are another way to express the likelihood of an event, related to probability.

• Actual odds against event A: $\frac{P(\bar{A})}{P(A)}$, usually expressed as a:b.

• Actual odds in favor of event A: $\frac{P(A)}{P(\bar{A})}$, the reciprocal of the odds against.

• Payoff odds against event A: (net profit):(amount bet)

Summary Table: Measures of Center and Variation

Measure	Definition	Formula	Resistant?
Mean	Arithmetic average of all values	$\bar{x} = \frac{\sum x}{n}$	No
Median	Middle value in ordered data	Sort data, find middle value	Yes
Mode	Most frequent value(s)	Value(s) with highest frequency	Yes
Midrange	Midpoint between max and min	$\frac{\text{Max} + \text{Min}}{2}$	No
Range	Difference between max and min	$\text{Max} - \text{Min}$	No
Standard Deviation	Average distance from mean	$s = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}$	No
Variance	Square of standard deviation	$s^2$	No
Mean Absolute Deviation	Mean of absolute deviations from mean	$\frac{\sum |x - \bar{x}|}{n}$	Yes

Key Vocabulary

• Measure of Center: Value at the center or middle of a data set.

• Resistant Statistic: Not significantly affected by outliers (e.g., median).

• Statistic: A numerical measurement describing a sample.

• Parameter: A numerical measurement describing a population.

• z Score: Number of standard deviations a value is from the mean.

• Percentile: Measure dividing data into 100 equal groups.

• Quartile: Measure dividing data into four equal groups.

Example: Suppose a data set has the values 2, 4, 6, 8, 10.

• Mean: $\bar{x} = \frac{2+4+6+8+10}{5} = 6$

• Median: 6 (middle value)

• Mode: No mode (all values occur once)

• Midrange: $\frac{10+2}{2} = 6$

• Range: $10 - 2 = 8$

• Standard deviation: $s = \sqrt{\frac{(2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2}{5-1}} = \sqrt{\frac{16+4+0+4+16}{4}} = \sqrt{10} \approx 3.16$

• z score for 10: $z = \frac{10-6}{3.16} \approx 1.27$

Additional info: Some formulas and explanations have been expanded for clarity and completeness. The notes include all key definitions, properties, and formulas relevant to measures of center and variation, as well as z scores, percentiles, quartiles, and odds, as covered in an introductory statistics course.