Cumulative Frequency, Mode, and Probability IB Math AI SL 4.2-4.6

Cumulative Frequency

• Cumulative frequency is a running total of occurrences less than a specific x value.

• To work with ranges of data, upper boundaries are used. These are plotted against the cumulative frequency.

Example: Calculating Cumulative Frequency

• Consider score value ranges:

  • 10-20

  • 20-30

  • 30-40

  • 40-50

  • 50-60

  • 60-70

  • 70-80

  • 80-90

  • 90-100

• Frequencies within each range are:

  • 10-20: 2

  • 20-30: 5

  • 30-40: 7

  • 40-50: 21

  • 50-60: 36

  • 60-70: 40

  • 70-80: 27

  • 80-90: 9

  • 90-100: 3

• Upper bounds for each range:

  • 20, 30, 40, 50, 60, 70, 80, 90, 100

• Cumulative frequency is calculated by adding up frequencies from previous ranges:

  • 2

  • 2 + 5 = 7

  • 7 + 7 = 14

  • 14 + 21 = 35

  • 35 + 36 = 71

  • 71 + 40 = 111

  • 111 + 27 = 138

  • 138 + 9 = 147

  • 147 + 3 = 150

Using Cumulative Frequency

• To find how many data points are less than a certain value, look at the cumulative frequency up to that value's range.

• For example, if asked how many points are less than 60, look at the cumulative frequency for the 50-60 range, which is 71.

Cumulative Frequency Chart

• A cumulative frequency chart is a visual representation of the data.

Finding Median and Quartiles

• For a data set of 150 points:

  • Median is at the 75th data point.

  • Quartile 1 is at the 37.5th data point.

  • Quartile 3 is at the 112.5th data point.

• To find these values on the chart:

  • Locate the corresponding cumulative frequency on the y-axis.

  • Trace horizontally to the curve.

  • Drop down to the x-axis to find the value.

• Calculations for Quartiles:

  • Quartile 1: Data Points * (1/4)

  • Median: Data Points * (1/2)

  • Quartile 3: Data Points * (3/4)

Percentiles

• To determine where a certain percentile falls (e.g., the 90th percentile), multiply the percentile by the total number of data points.

  • Example: 90th percentile is 0.9 * 150 = 135. Find 135 on the cumulative frequency chart to determine the corresponding value.

Mode and Modal Class

• Mode: The value that appears most often in a dataset.

• Modal Class: The range that shows up most frequently.

Example: Identifying Modal Class

• Given the frequencies for the ranges:

  • 10-20: 2

  • 20-30: 5

  • 30-40: 7

  • 40-50: 21

  • 50-60: 36

  • 60-70: 40

  • 70-80: 27

  • 80-90: 9

  • 90-100: 3

• The modal class is 60-70 because it has the highest frequency (40).

Standard Deviation

• Standard Deviation: The average difference of all points from the mean.

• Calculator Steps:

  • Menu -> Statistics -> One-Variable Calculation

• A small standard deviation indicates that data points are closely clustered around the mean.

• A large standard deviation indicates a wider spread of data points from the mean.

• Distance and Difference: The distance from a data point to the average.

Impact of Basic Operations on Standard Deviation and Mean

• If a basic operation (addition, subtraction) is performed on every data point, the spread (standard deviation) does not change, only the mean.

• Multiplication and division affect the mean.

Pearson's Correlation Coefficient

• Pearson's correlation coefficient (r) measures the linear correlation between two sets of data.

• It ranges between -1 and 1.

• Calculator Steps:

  • Name x and y columns.

  • Menu -> Statistics -> Linear Regression (choose fx=ax+b).

Linear Regression Line

• The calculator provides the linear regression line equation,y = mx + b, where m is the slope, and b is the y-intercept.

Probability

Key Vocabulary

• Experiment: A repeatable procedure with a set of possible outcomes.

• Sample Space: The set of all possible outcomes of an experiment.

• Event: A subset of the sample space.

Relative Frequency

• The number of times an event occurs divided by the total number of trials.

Basic Probability

• If all outcomes are equally likely and A is an event, the probability of A is the number of outcomes in A divided by the total number of possible outcomes.

Probability of an Event Not Happening

• Denoted as A'.

• P(A') = 1 - P(A)

Expected Occurrences

• E = n * p

• Where n is the number of trials and p is the probability of a specific event.

Example: expected occurrences

• If you roll a dice 12 times, what's probability of rolling less than 3?

• 12 * (1/3) = 4

• You should expect 4 rolls to be less than 3.

Venn Diagrams

• Visual representation of sets and their relationships, useful in probability.

Mutually Exclusive Events

• Events that cannot occur at the same time.

• P(A \cap B) = 0