Descriptive Statistics: Measures of Central Tendency and Distribution Shapes

Measures of Central Tendency

Definition and Importance

Measures of central tendency are statistical values that represent a typical or central entry of a data set. They are essential for summarizing and understanding the general behavior of data.

• Mean

• Median

• Mode

Mean (Average)

The mean is the sum of all data entries divided by the number of entries. It is a reliable measure because it incorporates every value in the data set.

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

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

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

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

• $\Sigma x = 1810$

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

Median

The median is the value that lies in the middle of an ordered data set, dividing it into two equal parts.

• If the number of entries is odd: the median is the middle entry.

• If the number of entries is even: the median is the mean of the two middle entries.

Example: For the ordered weights 223, 235, 235, 268, 274, 285, 290 (seven entries), the median is 268 pounds. If one entry is removed (six entries), the median is $(235 + 268)/2 = 251.5$ pounds.

Mode

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

• If no entry is repeated, there is no mode.

• If two entries share the highest frequency, the data set is bimodal.

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

Example (Categorical Data):

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

The mode is Democrat (highest frequency).

Comparing Mean, Median, and Mode

All three measures describe a typical entry, but each has advantages and disadvantages:

• Mean: Uses all data entries, but is sensitive to outliers.

• Median: Not affected by outliers; best for skewed data.

• Mode: Useful for categorical data; may not represent a typical value in numerical data.

Example: For the ages in a class:

Ages in a class
20 20 20 20 21 21 21 22 22 23 23 23 24 24 65

• Mean: $\bar{x} \approx 23.8$ years

• Median: $\frac{21 + 22}{2} = 21.5$ years

• Mode: 20 years

The mean is influenced by the outlier (65), while the median is not. The mode does not represent a typical entry.

Weighted Mean

Definition and Calculation

The weighted mean is used when data entries have different weights. It is calculated as:

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

Example: Calculating grade point average:

Final Grade	Credit Hours
C	3
D	4
A	1
C	2
B	3

Weighted mean: $\frac{40}{16} = 2.5$

Mean of Grouped Data

Mean of a Frequency Distribution

For grouped data, the mean is approximated using class midpoints and frequencies:

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

• $n = \Sigma f$

Steps:

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

2. Sum the products of midpoints and frequencies: $\Sigma xf$

3. Sum the frequencies: $n = \Sigma f$

4. Calculate the mean: $\bar{x} = \frac{\Sigma xf}{n}$

Example:

Class midpoint, x	Frequency, f	xf
172.5	3	517.5
208.5	2	417.0
240.5	5	1202.5
280.5	6	1683.0
316.5	7	2215.5
352.5	4	1410.0
388.5	3	1165.5
Total	30	8631.0

Mean: $\bar{x} = \frac{8631}{30} = 287.7$

Additional info: This is an estimate since it uses class midpoints.

The Shape of Distributions

Symmetric Distribution

A symmetric distribution has a vertical line through the center, creating mirror-image halves. Mean, median, and mode are typically equal.

Uniform Distribution (Rectangular)

In a uniform distribution, all entries or classes have equal or nearly equal frequencies. The distribution is symmetric.

Skewed Left Distribution (Negatively Skewed)

A skewed left distribution has a tail that extends more to the left. The mean is less than the median.

Skewed Right Distribution (Positively Skewed)

A skewed right distribution has a tail that extends more to the right. The mean is greater than the median.