Measures of Central Tendency and Distribution Shapes in Statistics

Measures of Central Tendency

Definitions

Measures of central tendency are statistical values that represent the center or typical value of a dataset. They are essential for summarizing and understanding data distributions.

• Sample: Any measurement taken from a sample (denoted by Roman letters).

• Population: Any measurement taken from a population (denoted by Greek letters).

• Frequency Distribution Table: Organizes raw data into table form; the frequency indicates the number of occurrences of any given data point.

Example Frequency Table:

x (Data Pt)	f (frequency)
0	1
1	2
2	3

$\sum f = 7$ (sample size)

Notation

Measure	Statistic (Sample)	Parameter (Population)
Mean	$\bar{x}$	$\mu$
Median	$m_{e}$	MD
Mode	$m_{o}$	MO
Midrange	$m_{r}$	MR

• $\sum x$: Sum of all data values x

• n: sample size

• f: frequency

Calculating Measures of Central Tendency

Mean

The mean is the arithmetic average of a dataset and is calculated as:

$\bar{x} = \frac{\sum x}{n}$

• Mean cannot be calculated for nominal data (qualitative data).

Median

The median is the middle value in a ranked dataset. If the dataset has an odd number of values, the median is the middle value. If even, it is the average of the two middle values.

• Rank data in order (low to high).

• Median position: $\frac{n+1}{2}$

Mode

The mode is the value that occurs most frequently in the dataset. A dataset may have one mode (unimodal), more than one mode (multimodal), or no mode.

• Unimodal: One value occurs most often.

• Bimodal: Two values occur most often.

• No mode: All values occur with equal frequency.

Midrange

The midrange is the average of the lowest and highest values in a dataset:

$m_{r} = \frac{\text{min} + \text{max}}{2}$

Rounding Rule

• When calculating measures of central tendency, express the answer with one additional decimal place than the raw data.

Electronic Tools

• For tabulated data, use frequency tables and calculators/statistical software to compute mean, median, mode, and midrange.

• Calculator steps: Enter data and frequencies, then use statistical functions to compute measures.

Raw Data and Outliers

Raw Data Example

Sample data: 3.7, 4.8, 4.8, 4.0, 4.7, 5.1, 4.3

• Mean: $\bar{x} = 4.49$

• Median: $m_{e} = 4.7$

• Mode: $m_{o} = 4.8$

• Midrange: $m_{r} = \frac{3.7 + 5.1}{2} = 4.4$

Raw Data Example with Outlier

Sample data: 3.7, 4.8, 4.8, 4.0, 4.7, 8.1, 4.3

• Mean: $\bar{x} = 4.91$

• Median: $m_{e} = 4.7$

Outliers can cause the mean and median to be significantly different. The mean is more sensitive to outliers than the median or mode.

• If the outlier is high, the mean will be higher than the median.

• If the outlier is low, the mean will be lower than the median.

Trimmed Mean: To mitigate the effects of outliers, a trimmed mean can be used, where a certain percentage of the lowest and highest data values are removed before calculating the mean.

Frequency Data Examples

Frequency Table Example

Number of books read by 19 students:

x (books)	f (students)	f*x	c.f. (cumulative freq.)
0	2	0	2
1	3	3	5
2	5	10	10
3	4	12	14
4	5	20	19

• Mean: $\bar{x} = \frac{51}{19} = 2.7$

• Median: Median position is $\frac{n+1}{2} = 10$th value; median is 2.

• Mode: 4 (most frequent value)

• Midrange: $\frac{0 + 4}{2} = 2$

Frequency Data Example Using Classes

Endurance times for 80 students:

Class Bounds (hrs)	f (frequency)	x (midpt of class)	f*x	c.f.
52.5 - 63.5	6	58	348	6
63.5 - 74.5	12	69	828	18
74.5 - 85.5	25	80	2000	43
85.5 - 96.5	18	91	1638	61
96.5 - 107.5	14	102	1428	75
107.5 - 118.5	5	113	565	80

• Mean: $\bar{x} = \frac{6807}{80} = 85.1$

• Median: Median position is $\frac{n+1}{2} = 40.5$th value; median is 80.

• Mode: 80 (most frequent class midpoint)

• Midrange: $\frac{58 + 113}{2} = 85.5$

Frequency Table for Median and Mode

x (Data Value)	f (Frequency)	c.f. (cumulative)
8	112	112
17	102	214
27	197	411
37	520	931
43	186	1217

• Median: $n = 1219$; median position is $\frac{n+1}{2} = 610$th value; median is 37.

• Mode: 37 (largest frequency, 520 occurrences)

Weighted Mean

Definition and Calculation

Weighted means are used when not all values are equally represented. The formula is:

$\bar{x} = \frac{\sum w x}{\sum w}$

• Weights (w) represent the relative importance or frequency of each value.

Weighted Mean Example: Stock Portfolio

Price, x	Stocks, w	Product, w x
10	8	80
12	20	240
17	15	255
20	30	600
35	7	245

• Weighted mean: $\bar{x} = \frac{80 + 240 + 255 + 600 + 245}{80} = 23.88$

Weighted Mean Example: Camera Ratings

Category	w (weight)	Cony X (score)	Cony w x	Sanon X (score)	Sanon w x
Image Quality	0.5	8	4	9	4.5
Battery Life	0.3	6	1.8	6	1.8
Zoom Range	0.2	7	1.4	6	1.2

• Cony weighted mean: $\bar{x} = \frac{4 + 1.8 + 1.4}{1} = 7.2$

• Sanon weighted mean: $\bar{x} = \frac{4.5 + 1.8 + 1.2}{1} = 6.9$

Weighted Mean Example: GPA Calculation

Letter Grade	Numeric Grade (x)	Number of Classes	Credit per class (w)	Total Credits	Total Points Earned (w x)
A	4	1	4	4	16
B	3	3	3	9	27
C	2	1	3	3	6
D	1	1	4	4	4

• Weighted mean (GPA): $\bar{x} = \frac{16 + 27 + 6 + 4}{20} = 2.65$

Shapes of Distributions

Symmetric Distributions

When a vertical line can be drawn through the middle of the graph and the resulting halves are approximately mirror images, the distribution is symmetric.

• Mode = Mean = Median

Uniform Distribution

All data values have the same (or nearly the same) frequency. The graph is rectangular.

Skewed Distributions

• Right-skewed (Positively Skewed): Disproportionately large amount of small values; long tail on right. Outliers at high values. Mean > Median > Mode.

• Left-skewed (Negatively Skewed): Disproportionately large amount of large values; long tail on left. Outliers at low values. Mean < Median < Mode.

Graphical Examples

• Uniform: Bar graph with equal heights.

• Symmetric (Normal): Bell-shaped curve.

• Positively Skewed: Tail extends to the right.

• Negatively Skewed: Tail extends to the left.

Summary Table: Measures of Central Tendency

Measure	Definition	Formula	Sensitivity to Outliers
Mean	Arithmetic average	$\bar{x} = \frac{\sum x}{n}$	High
Median	Middle value	Middle position in ranked data	Low
Mode	Most frequent value	Value with highest frequency	Low
Midrange	Average of min and max	$m_{r} = \frac{\text{min} + \text{max}}{2}$	High
Weighted Mean	Mean with weights	$\bar{x} = \frac{\sum w x}{\sum w}$	Depends on weights

Key Points

• Central tendency measures summarize the center of a dataset.

• Mean is sensitive to outliers; median and mode are more robust.

• Weighted mean accounts for varying importance or frequency of data points.

• Distribution shape affects the relationship between mean, median, and mode.