Geometry, Measurement & Data

Data Analysis and Probability

Grade 7 · Grade 8 · Grade 9

✓By the end of this lesson students will be able to calculate and interpret measures of central tendency (mean, median, mode) and spread (range) for a given set of data.
✓By the end of this lesson students will be able to determine the theoretical probability of an event occurring.
✓By the end of this lesson students will be able to determine the experimental probability of an event occurring based on collected data.
✓By the end of this lesson students will be able to compare and contrast theoretical and experimental probabilities.

Key concepts

Measures of Central Tendency

Measures of central tendency describe the 'centre' or typical value of a data set. The three main measures are the mean, median, and mode.

Mean (Average)

The mean is calculated by adding all the values in a data set and then dividing by the number of values. It is often represented by the symbol ōX (read as 'x-bar').

Mean = (Sum of all values) / (Number of values)

Median

The median is the middle value in a data set when the values are arranged in numerical order. If there is an even number of values, the median is the average of the two middle values.

Mode

The mode is the value that appears most frequently in a data set. A data set can have one mode (unimodal), more than one mode (multimodal), or no mode if all values appear with the same frequency.

Range

The range is a measure of spread that describes the difference between the highest and lowest values in a data set. It indicates how spread out the data is.

Range = Highest value - Lowest value

Theoretical Probability

Theoretical probability is the likelihood of an event occurring based on reasoning and mathematical calculations, assuming all outcomes are equally likely. It's what we expect to happen.

P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes)

Sample Space

The sample space is the set of all possible outcomes of a probability experiment.

Experimental Probability

Experimental probability is the likelihood of an event occurring based on the results of an actual experiment or observation. It's what actually happens when you conduct trials.

P(Event) = (Number of times the event occurs) / (Total number of trials)

Key facts to remember

1The mean is the average of a data set.
2The median is the middle value when data is ordered.
3The mode is the most frequent value in a data set.
4The range is the difference between the highest and lowest values.
5Theoretical probability is based on what should happen.
6Experimental probability is based on what actually happens in trials.
7Probability values always fall between 0 and 1 (inclusive), or 0% and 100%.
8P(Event) = (Number of favourable outcomes) / (Total number of outcomes).

Worked examples

Example 1

A Grade 8 math class had the following test scores: 78, 85, 92, 78, 65, 88, 90, 72, 85, 80. Calculate the mean, median, mode, and range of these scores.

IStep 1: Order the data from least to greatest: 65, 72, 78, 78, 80, 85, 85, 88, 90, 92.

IIStep 2: Calculate the Mean (Sum of values / Number of values):\n (65 + 72 + 78 + 78 + 80 + 85 + 85 + 88 + 90 + 92) / 10\n = 793 / 10\n = 79.3

IIIStep 3: Calculate the Median (Middle value):\n Since there are 10 values (an even number), the median is the average of the 5th and 6th values.\n Ordered data: 65, 72, 78, 78, *80, 85*, 85, 88, 90, 92\n Median = (80 + 85) / 2\n = 165 / 2\n = 82.5

IVStep 4: Calculate the Mode (Most frequent value):\n The values 78 and 85 both appear twice, which is more than any other value.\n Mode = 78 and 85 (bimodal)

VStep 5: Calculate the Range (Highest value - Lowest value):\n Range = 92 - 65\n = 27

Answer

Mean = 79.3, Median = 82.5, Mode = 78 and 85, Range = 27

Always order your data first when finding the median or range to avoid errors.

Example 2

A standard six-sided die is rolled. What is the theoretical probability of:\n a) Rolling an even number?\n b) Rolling a number greater than 4?\n c) Rolling a 7?

IStep 1: Identify the sample space (all possible outcomes).\n Sample Space = {1, 2, 3, 4, 5, 6}. Total number of possible outcomes = 6.

IIStep 2a: Identify favourable outcomes for rolling an even number.\n Favourable outcomes = {2, 4, 6}. Number of favourable outcomes = 3.\n P(Even number) = (Number of favourable outcomes) / (Total number of possible outcomes)\n P(Even number) = 3 / 6 = 1/2

IIIStep 2b: Identify favourable outcomes for rolling a number greater than 4.\n Favourable outcomes = {5, 6}. Number of favourable outcomes = 2.\n P(Number > 4) = (Number of favourable outcomes) / (Total number of possible outcomes)\n P(Number > 4) = 2 / 6 = 1/3

IVStep 2c: Identify favourable outcomes for rolling a 7.\n Favourable outcomes = {}. Number of favourable outcomes = 0.\n P(Rolling a 7) = 0 / 6 = 0

Answer

a) P(Even number) = 1/2\nb) P(Number > 4) = 1/3\nc) P(Rolling a 7) = 0

Probabilities can be expressed as fractions, decimals, or percentages. A probability of 0 means an event is impossible, and a probability of 1 means an event is certain.

Example 3

A coin is flipped 50 times. It lands on heads 28 times and tails 22 times. \n a) What is the experimental probability of flipping heads?\n b) What is the experimental probability of flipping tails?\n c) How do these compare to the theoretical probabilities?

IStep 1: Determine the total number of trials.\n Total trials = 50.

IIStep 2a: Calculate the experimental probability of flipping heads.\n Number of times heads occurred = 28.\n P(Heads) = (Number of times heads occurs) / (Total number of trials)\n P(Heads) = 28 / 50 = 14 / 25 = 0.56 or 56%

IIIStep 2b: Calculate the experimental probability of flipping tails.\n Number of times tails occurred = 22.\n P(Tails) = (Number of times tails occurs) / (Total number of trials)\n P(Tails) = 22 / 50 = 11 / 25 = 0.44 or 44%

IVStep 3c: Compare to theoretical probabilities.\n For a fair coin, the theoretical probability of flipping heads is 1/2 or 0.5 (50%).\n The theoretical probability of flipping tails is also 1/2 or 0.5 (50%).\n The experimental probability of heads (0.56) is slightly higher than the theoretical (0.5).\n The experimental probability of tails (0.44) is slightly lower than the theoretical (0.5).

Answer

a) P(Heads) = 14/25 or 0.56\nb) P(Tails) = 11/25 or 0.44\nc) The experimental probabilities (0.56 for heads, 0.44 for tails) are close to, but not exactly the same as, the theoretical probabilities (0.5 for heads, 0.5 for tails).

As the number of trials increases, experimental probability generally gets closer to theoretical probability (Law of Large Numbers).

Common mistakes

✗Forgetting to order the data before finding the median.
✗Confusing mean, median, and mode.
✗Incorrectly calculating the median for an even number of data points (not averaging the two middle values).
✗Expressing probability as a number greater than 1 or less than 0.
✗Mixing up the numerator and denominator in probability calculations (e.g., total outcomes / favourable outcomes).

Exam tips

★Always show your work, especially for mean and range calculations.
★Read the question carefully to determine if theoretical or experimental probability is required.
★Simplify probability fractions to their lowest terms unless otherwise specified.
★For probability questions, clearly state the sample space and favourable outcomes.

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