Frequency Table & Distribution Analysis

• Study notes for analyzing data in frequency tables and distributions

Frequency Table Analysis

• Frequency (f) indicates how often a value appears in a dataset.
• Cumulative frequency is calculated by adding each frequency to the sum of the frequencies before it.
• Range is the difference between the maximum and minimum values in a dataset.
• Median is the middle value in a dataset, which can be found using cumulative frequencies.
• Mode is the value that appears most frequently in a dataset.
• Q1 and Q3 represent the 25th and 75th percentiles of the data, respectively.
• Box-and-whisker plots are created using the minimum value, Q1, median, Q3, and maximum value to visualize the distribution.

Measures of Central Tendency

• Mean (x̄) is calculated by summing all values multiplied by their frequencies and then dividing by the total number of values.
• The formula for the mean is x̄ = (Σfx) / n, where f is frequency, x is the data value, fx is f × x, and n is the total frequency.

Shape of Distribution

• Symmetry in a distribution indicates a normal distribution.
• Skewed Right distributions have a tail extending to the right.
• Skewed Left distributions have a tail extending to the left.
• If the mode, median, and mean are all different, the distribution is likely skewed.

Standard Deviation

• Standard deviation measures the spread of data around the mean.
• To calculate it, find the mean, subtract the mean from each value, square the results, find the average of the squared deviations, and then take the square root.
• The formula for standard deviation (σ) is σ = √[(Σ(x - x̄)²) / n].

Z-Score

• Z-score indicates how many standard deviations an individual data point is from the mean.
• The formula for calculating the z-score is z = (x - μ) / σ, where x is the individual score, μ is the mean, and σ is the standard deviation.
• Usual z-scores fall between -2 and +2.
• Unusual z-scores are less than -2 or greater than +2.

Relative Comparison

• Z-scores are used to compare different data points by determining their relative positions within their respective distributions.

Finding the Range

• Range is determined by subtracting the lowest value from the highest value in the dataset.

Finding the Mode

• The mode is the value that appears most often in the dataset.

Finding the Median

• Cumulative frequency is used to find the middle student (n ÷ 2) where n = total number of students.
• Locate which time interval that middle student falls into

Finding the Mean

• Multiply each time value × its frequency
• Add them all up
• Divide by the total number of students.
• The formula is: Mean = (Sum of all (value × frequency)) ÷ (Total number of students)

Finding Q1 and Q3

• Q1 is the median of the lower half of the data (first 25% of students).
• Q3 is the median of the upper half of the data (first 75% of students).
• After finding the overall median, split the data into lower and upper halves and find the median of each.

Shape of Distribution

• Symmetrical distributions resemble a bell curve.
• Skewed left distributions have a long tail on the left (mean < median).
• Skewed right distributions have a long tail on the right (mean > median).

Box-and-Whisker Plot

• Draw a number line that covers all your data
• Plot: Minimum (smallest data point), Q1 (lower quartile), Median (middle), Q3 (upper quartile), Maximum (largest data point)
• Draw a box from Q1 to Q3, and a line at the median.
• Extend “whiskers” from the box to the min and max.

Standard Deviation

• Subtract the mean (x - mean) = deviation and Square the deviation ((x - mean)^2)
• Standard deviation = sqrt[(sum of squares) ÷ (n-1)] (because it’s a sample, not a population.)

Memory Tricks

• Mean = Average (total ÷ how many)
• Median = Middle (when ordered)
• Mode = Most (most frequent)
• Range = Difference (biggest - smallest)
• Standard deviation = How spread out the numbers are