Measures of Central Tendency and the Shape of Distributions

Measures of Central Tendency

Definition and Importance

Measures of central tendency are values that represent a typical or central entry of a data set. They are essential for summarizing and describing the main features of a collection of data.

• Mean

• Median

• Mode

Mean (Average)

• The mean is the sum of all the data entries divided by the number of entries.

• Sigma notation: $\sum x$ denotes the sum of all data entries $x$ in the data set.

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

• Sample mean: $\bar{x} = \frac{\sum x}{n}$

Example: For the weights 274, 235, 223, 268, 290, 285, 235:

• $\sum x = 1810$

• $n = 7$

• $\bar{x} = \frac{1810}{7} \approx 258.6$ pounds

Median

• The median is the value that lies in the middle of the data when the data set is ordered.

• It divides the data set into two equal parts.

• If the data set has an odd number of entries: the median is the middle entry.

• If the data set has an even number of entries: the median is the mean of the two middle entries.

Example: For the ordered weights 223, 235, 235, 268, 274, 285, 290, the median is 268 pounds.

Example (even entries): For 223, 235, 235, 268, 274, 290, the median is $\frac{235 + 268}{2} = 251.5$ pounds.

Mode

• The mode is the data entry that occurs with the greatest frequency.

• Unimodal: One entry occurs most frequently.

• Bimodal: Two entries occur with the same greatest frequency.

• Multimodal: Three or more entries occur with the same greatest frequency.

• If no entry is repeated or more than three entries occur with the same greatest frequency, there is no mode.

Example: For the weights 223, 235, 235, 268, 274, 285, 290, the mode is 235 pounds.

Political Party	Frequency, f
Democrat	46
Republican	34
Independent	39
Other/don't know	5

The mode is Democrat (highest frequency).

Comparing the Mean, Median, and Mode

• All three measures describe a typical entry of a data set.

• Advantage of the mean: It uses every entry in the data set, making it reliable.

• Disadvantage of the mean: It is greatly affected by outliers (values far removed from the rest).

Example: Ages in a class: Mean = 23.8, Median = 20.5, Mode = 20. The outlier (65) skews the mean.

Graphical comparison can help determine which measure best represents the data. In the example, the median best describes the data set.

When to Use Mean vs. Median

• Use the mean: When data are fairly symmetric and have no outliers.

• Use the median: When data are skewed or contain outliers.

Weighted Mean

• The mean of a data set whose entries have varying weights.

• Formula: $\bar{x} = \frac{\sum xw}{\sum w}$, where $w$ is the weight of each entry $x$.

Example: Mark's GPA calculation:

Points, x	Credit hours, w	xw
2	3	6
2	4	8
1	1	1
4	3	12
2	2	4
3	3	9
Total	16	40

• $\bar{x} = \frac{40}{16} = 2.5$

Mean of Grouped Data

• For a frequency distribution, the mean is approximated by:

$\bar{x} = \frac{\sum xf}{n}$, where $x$ is the class midpoint and $f$ is the frequency.

$n = \sum f$

• Steps:

  1. Find the midpoint of each class: $x = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

  2. Find the sum of the products of the midpoints and the frequencies: $\sum xf$

  3. Find the sum of the frequencies: $n = \sum f$

  4. Find the mean: $\bar{x} = \frac{\sum xf}{n}$

Example: Screen time for 30 adults:

Class	Frequency, f
155-190	3
191-226	2
227-262	5
263-298	6
299-334	7
335-370	4
371-406	3

Estimated mean: $\bar{x} = \frac{8631}{30} = 287.7$ minutes

Additional info: This is an estimate because it uses class midpoints, not the original data.

The Shape of Distributions

Symmetric Distribution

• A vertical line can be drawn through the middle of the graph, and the resulting halves are approximately mirror images.

Uniform Distribution (Rectangular)

• All entries or classes have equal or approximately equal frequencies.

• Symmetric.

Normal Distribution

• A distribution that is both symmetrical and unimodal.

• The mean, median, and mode are equal.

Skewed Left Distribution (Negatively Skewed)

• The "tail" of the graph elongates more to the left.

• The mean is to the left of the median: $\text{Mean} < \text{Median}$

Skewed Right Distribution (Positively Skewed)

• The "tail" of the graph elongates more to the right.

• The mean is to the right of the median: $\text{Mean} > \text{Median}$