Statistics: Frequency Distributions

Frequency Distributions and Their Graphs

• A frequency distribution is a table that organizes data into classes or intervals, showing the count of data entries in each class.
• The frequency (f) of a class is the number of data entries in that class.
• Class width is the difference between the lower and upper class limits.
• To construct a frequency distribution, first decide on the number of classes (usually between 5 and 20).
• Then find the class width by determining the range of the data and dividing by the number of classes. Round up to the nearest convenient number.
• Find the class limits. Starting with the minimum, add class width. Find upper class limits similarly.
• Create a tally mark for each data entry in the appropriate class.
• Count the tally marks to get the frequency for each class.
• Relative frequency is the proportion of data in a class (class frequency divided by total sample size).

Constructing a Frequency Distribution

• The number of classes will determine how to evenly organize data.
• Find class width by determining range and dividing by the number of classes and rounding up.
• Find class limits (upper and lower) by using class width and adding it to the lower class limit of the preceding class.

Graphs of Frequency Distributions

• A histogram is a bar graph that shows the frequency distribution of data.
• The horizontal axis displays the data values (numerical), while the vertical axis shows the frequencies.
• The bars must touch and show the frequencies of each class.
• A frequency polygon is a line graph showing the frequency distribution of data.
• The points are plotted at each class midpoint and the points are joined together.
• Relative frequency histograms use relative frequencies instead of frequencies on the vertical axis.
• Ogives are line graphs that show cumulative frequencies.
• Plot points using upper class boundaries and their corresponding cumulative frequencies.
• Connect points from left to right using upper class boundaries.

Class Boundaries

• Class boundaries are numbers that separate classes without gaps.
• Find the upper and lower class boundary by taking the upper class limit minus 0.5 and lower class limit + 0.5.

Stem-and-Leaf Plot

• Each data value is separated into a stem and a leaf.
• The stem contains the leading digits, and the leaf contains the trailing digits.
• Similar to a histogram, but it contains all the original data values.

Dot Plot

• A dot plot uses dots above a number line or horizontal axis to represent the data values.
• Each dot corresponds to a data entry.
• If a data entry is repeated, multiple dots are placed above the same value on the axis.

Pie Chart

• A pie chart is a circle divided into sectors.
• The size of each sector corresponds to the proportion or percentage of the frequency of each category.
• Relative frequency (percentage) for each category is multiplied by 360 degrees to calculate the sector size.

Measures of Central Tendency

• Mean: The sum of all entries divided by the number of entries. (average)
• Median: The middle value in an ordered data set.
• Mode: The value that appears most frequently in a data set.

Measures of Variation

• Range: The difference between the maximum and minimum values in a data set.
• Variance: An average of the squared deviations from the mean in a data set (population variance and sample variance formulas different).
• Standard Deviation: The square root of the variance (population standard deviation and sample standard deviation formulas different).

Weighted Mean

• The mean of a dataset where each entry has a weight.
• Weights of entries are multiplied by the entries and added up, and this sum is divided by the total of all weights.

Percentiles

• Percentiles divide data into 100 equal parts.
• The pth percentile is a value such that p percent of the data values are less than or equal to that value.
• A percentile is a position in a sorted dataset.

Quartiles

• Quartiles are percentiles that divide data into 4 parts.
• Q1 is the first quartile, Q2 is the median, and Q3 is the third quartile.
• Interquartile range (IQR): the difference between the third and first quartiles.

Box-and-Whisker Plot

• A box-and-whisker plot displays the five-number summary of a data set.
• Minimum value, first quartile, median, third quartile, and maximum value.
• A box is drawn using Q1, Q3 and a vertical line at the median Q2.
• Whiskers extend from the box to the minimum and maximum.

Standard Scores (z-scores)

• A z-score represents the number of standard deviations a data point is from the mean.
• z=(x-mean)/standard deviation.