Descriptive Statistics: Frequency Distributions, Graphs, and Measures of Central Tendency, Variation, and Position

Section 2.1 - Frequency Distributions and Their Graphs

Frequency Distributions

A frequency distribution is a table that organizes data into classes or intervals, showing how many data entries fall into each class.

• Class Limits: Each class has a lower limit (smallest value that can belong) and an upper limit (largest value that can belong).

• Class Width: The distance between consecutive lower (or upper) limits.

• Range: The difference between the maximum and minimum data entries.

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

Constructing a Frequency Distribution

1. Decide on the number of classes (usually 5–20 for clarity).

2. Find the class width:

   1. Determine the range.

   2. Divide the range by the number of classes.

   3. Round up to the next convenient number.

3. Find class limits:

   • Start with the minimum data entry as the first lower limit.

   • Calculate remaining lower and upper limits, ensuring classes do not overlap.

4. Tally each data entry in the appropriate class.

5. Count tallies to find the frequency for each class.

Other Key Terms

• Midpoint: The value halfway between the lower and upper class limits.

  • Formula: $\text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

• Relative Frequency: The proportion or percentage of data in a class.

  • Formula: $\text{Relative Frequency} = \frac{f}{n}$, where $f$ is class frequency and $n$ is sample size.

• Cumulative Frequency: The sum of the frequencies for a class and all previous classes. The cumulative frequency of the last class equals the sample size $n$.

Graphical Representations

• Frequency Histogram: A bar graph representing the frequency distribution. The horizontal axis is quantitative (data values), and the vertical axis shows frequencies. Bars touch to indicate continuous data, and start/end at class boundaries.

• Frequency Polygon: A line graph using the same scales as the histogram, emphasizing continuous change. Begins and ends on the horizontal axis.

• Relative Frequency Histogram: Similar to a frequency histogram, but the vertical axis shows relative frequencies.

• Cumulative Frequency Graph (Ogive): A line graph displaying cumulative frequencies at each upper class boundary. Starts at the lower boundary of the first class (cumulative frequency zero) and ends at the upper boundary of the last class (cumulative frequency equals sample size).

Section 2.2 - More Graphs and Displays

Stem-and-Leaf Plot

A stem-and-leaf plot separates each data value into a "stem" (all but the final digit) and a "leaf" (the final digit). It is similar to a histogram but retains the original data values.

• Example: For the number 47, the stem is 4 and the leaf is 7.

Dot Plot

A dot plot displays each data entry as a point above a horizontal axis, showing frequency by the number of dots at each value.

Pie Chart

A pie chart is a circular graph divided into sectors, each representing a category's proportion of the whole.

• The area of each sector is proportional to the category's frequency.

• To find the central angle for a sector:

  • Formula: $\text{Central Angle} = 360^\circ \times \text{Relative Frequency}$

• Example: If a category has 1,037 out of 4,072 entries, relative frequency is $\frac{1037}{4072}$, and the central angle is $360^\circ \times \frac{1037}{4072}$.

Pareto Chart

A Pareto chart is a vertical bar graph where bars represent frequency or relative frequency, arranged in decreasing order from left to right.

Paired Data Sets and Scatter Plots

A paired data set consists of two related data sets, with each entry in one corresponding to an entry in the other. A scatter plot graphs ordered pairs to show the relationship between two quantitative variables.

Time Series and Time Series Chart

A time series is a data set of quantitative entries taken at regular intervals over time. A time series chart graphs these values to show trends over time.

Section 2.3 - Measures of Central Tendency

Definition and Types

Measures of central tendency are values that represent a typical or central entry in a data set. The most common are the mean, median, and mode.

• Mean (Arithmetic Average): The sum of all data entries divided by the number of entries.

  • Formula: $\bar{x} = \frac{\sum x}{n}$ (sample mean)

  • Advantage: Uses all data entries; reliable.

  • Disadvantage: Sensitive to outliers.

• Median: The middle value in an ordered data set.

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

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

• Mode: The data entry that occurs most frequently.

  • No mode if all entries are unique.

  • More than one mode if multiple entries share the highest frequency.

• Weighted Mean: Used when data entries have different weights.

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

• Mean of a Frequency Distribution: Approximated by:

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

Distribution Shapes and Central Tendency

• Symmetric Distribution: Mean, median, and mode are equal; halves are mirror images.

• Uniform Distribution: All classes have equal (or nearly equal) frequencies; mean and median are the same.

• Skewed Left (Negatively Skewed): Tail extends left; mean is less than median.

• Skewed Right (Positively Skewed): Tail extends right; mean is greater than median.

Section 2.4 - Measures of Variation

Range

The range is the simplest measure of variation, calculated as the difference between the maximum and minimum data entries.

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

Deviation

• For a population: $\text{Deviation of } x = x - \mu$

• For a sample: $\text{Deviation of } x = x - \bar{x}$

Variance and Standard Deviation

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

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

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

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

• Interpretation: Standard deviation measures how spread out data are from the mean. $\sigma = 0$ means no variation; higher $\sigma$ means more spread.

Chebyshev’s Theorem

For any data set (regardless of distribution), at least $1 - \frac{1}{k^2}$ of the data lie within $k$ standard deviations of the mean ($k > 1$).

• Formula: $1 - \frac{1}{k^2}$

Sample Standard Deviation for Frequency Distributions

• Estimate using class midpoints ($x$) and frequencies ($f$):

  • Formula: $s = \sqrt{\frac{\sum (x - \bar{x})^2 f}{n - 1}}$, where $n = \sum f$

Coefficient of Variation (CV)

Describes the standard deviation as a percent of the mean.

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

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

Section 2.5 - Measures of Position

Fractiles

Fractiles are numbers that divide an ordered data set into equal parts.

• Quartiles: Divide data into four equal parts.

  • First quartile ($Q_1$): About 25% of data fall below or at $Q_1$.

  • Second quartile ($Q_2$): About 50% (the median).

  • Third quartile ($Q_3$): About 75% fall below or at $Q_3$.

• Deciles: Divide data into 10 equal parts.

• Percentiles: Divide data into 100 equal parts.

Interquartile Range (IQR)

The interquartile range measures the spread of the middle 50% of data.

• Formula: $IQR = Q_3 - Q_1$

Identifying Outliers Using IQR

1. Find $Q_1$ and $Q_3$.

2. Compute $IQR = Q_3 - Q_1$.

3. Multiply $IQR$ by 1.5.

4. Any data entry less than $Q_1 - 1.5(IQR)$ or greater than $Q_3 + 1.5(IQR)$ is an outlier.

Box-and-Whisker Plot

A box-and-whisker plot visually displays the minimum, $Q_1$, median ($Q_2$), $Q_3$, and maximum of a data set.

Percentile Rank

To find the percentile corresponding to a data entry $x$:

• Formula: $\text{Percentile of } x = \frac{\text{Number of data entries less than } x}{\text{Total number of data entries}} \times 100$

Standard Score (z-score)

The z-score indicates how many standard deviations a value $x$ is from the mean $\mu$.

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