Statistical Symbols and Calculator Output Interpretation (Chapters 1–3)

Statistical Symbols and Their Use in Descriptive Statistics

Population vs. Sample Notation

In statistics, it is essential to distinguish between data collected from a population and a sample. The symbols used for statistical measures differ depending on the source of the data, even though the calculation methods for some measures are similar.

• Population Mean (\( \mu \)): Used when data represents the entire population.

• Sample Mean (\( \overline{x} \)): Used when data is from a sample.

• The formula for both means is the same: $\overline{x} = \mu = \frac{\sum x}{n}$

• The numerical value for the mean does not change based on the symbol, only the context does.

Standard Deviation: Sample vs. Population

The standard deviation measures the spread of data values. The formulas differ for a sample and a population, resulting in different numerical values:

• Sample Standard Deviation (\( s \)): Used for sample data.

• Population Standard Deviation (\( \sigma \)): Used for population data.

• The formulas are:

Sample Standard Deviation: $s = \sqrt{\frac{\sum (x - \overline{x})^2}{n-1}}$

Population Standard Deviation: $\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$

• Because the denominators differ (\( n-1 \) for sample, \( N \) for population), the values of \( s \) and \( \sigma \) are generally not the same.

Common Statistical Symbols (Chapters 1–3)

The following table summarizes key symbols used in introductory statistics:

Symbol	Meaning
f	Frequency
\( \sum x \)	Summation of x-values
n	Sample size
N	Population size
\( \overline{x} \)	Sample mean
\( \mu \)	Population mean
s	Sample standard deviation
\( \sigma \)	Population standard deviation
\( s^2 \)	Sample variance
\( \sigma^2 \)	Population variance
Q1	First quartile
Q2	Median (second quartile)
Q3	Third quartile
IQR	Interquartile range (\( Q_3 - Q_1 \))
Pk	kth percentile

Interpreting TI-84 Calculator Output for 1-Variable Statistics

Understanding the TI-84 1-Var Stats Screen

The TI-84 calculator provides a summary of descriptive statistics for a single variable. The screen displays key values that correspond to the statistical symbols discussed above.

• \( \overline{x} \): Sample mean

• \( \sum x \): Sum of all data values

• \( \sum x^2 \): Sum of the squares of all data values

• Sx: Sample standard deviation (\( s \))

• \( \sigma x \): Population standard deviation (\( \sigma \))

• n: Number of data values (sample size)

• minX: Minimum data value

• Q1: First quartile

Example: The following TI-84 output summarizes a data set of 18 values:

• Sample mean (\( \overline{x} \)): 33.22

• Sum of values (\( \sum x \)): 598

• Sample standard deviation (Sx): 15.94

• Population standard deviation (\( \sigma x \)): 15.49

• Sample size (n): 18

• Minimum value: 12

• First quartile (Q1): 19

This output helps you quickly identify key descriptive statistics for further analysis or reporting.

Additional info: The TI-84 calculator is widely used in introductory statistics courses for computing and interpreting basic descriptive statistics. Understanding the correspondence between calculator output and statistical notation is essential for accurate reporting and interpretation.