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Statistics and Probability - Univariate Data and Distributions

Grade 7IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Univariate Data refers to data that consists of observations on only a single characteristic or attribute, such as the heights of students in a class.

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Quantitative Data is numerical and can be measured. It is divided into Discrete data (countable values like the number of pets) and Continuous data (measurable values like height or time, often involving decimals).

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Qualitative Data is categorical and describes qualities or characteristics, such as hair color or favorite sport.

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The Mean is the arithmetic average, calculated by xˉ=∑xn\bar{x} = \frac{\sum x}{n}.

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The Median is the middle value of a data set when the values are arranged in ascending or descending order. If the number of observations nn is odd, the median is the middle term. If nn is even, it is the average of the two middle terms.

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The Mode is the value that occurs most frequently in the data set. A set can have no mode, one mode, or be bimodal.

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The Range is a measure of spread, calculated as the difference between the highest and lowest values: Range=xmax−xminRange = x_{max} - x_{min}.

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Frequency Tables are used to organize data by listing each data value or category alongside the number of times it occurs (ff).

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Probability is the likelihood of an event occurring, expressed as a value between 00 and 11. P(E)=0P(E) = 0 means the event is impossible, and P(E)=1P(E) = 1 means it is certain.

📐Formulae

Mean (xˉ)=Sum of all observationsTotal number of observations=∑xn\text{Mean } (\bar{x}) = \frac{\text{Sum of all observations}}{\text{Total number of observations}} = \frac{\sum x}{n}

Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}

Position of Median=n+12th term\text{Position of Median} = \frac{n + 1}{2} \text{th term}

P(Event)=Number of favorable outcomesTotal number of possible outcomesP(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

💡Examples

Problem 1:

Find the mean, median, mode, and range of the following data set representing the number of goals scored by a soccer team in 7 matches: 2,1,4,1,3,0,32, 1, 4, 1, 3, 0, 3.

Solution:

First, we sum the values for the mean: 214130+314\begin{array}{r} 2 \\ 1 \\ 4 \\ 1 \\ 3 \\ 0 \\ + 3 \\ \hline 14 \end{array} Mean: xˉ=147=2\bar{x} = \frac{14}{7} = 2 goals. Next, we order the data: 0,1,1,2,3,3,40, 1, 1, 2, 3, 3, 4. Median: The middle (4th4^{th}) value is 22. Mode: The values 11 and 33 both appear twice, so the data is bimodal: 11 and 33. Range: 4−0=44 - 0 = 4.

Explanation:

To find the mean, we divide the sum of values by the count. For the median, we must arrange the numbers in order. The range is the gap between the largest and smallest numbers.

Problem 2:

A bag contains 55 red marbles, 33 blue marbles, and 22 green marbles. If one marble is picked at random, what is the probability P(Blue)P(\text{Blue})?

Solution:

Total number of marbles n=5+3+2=10n = 5 + 3 + 2 = 10. Number of favorable outcomes (blue marbles) =3= 3. P(Blue)=310=0.3P(\text{Blue}) = \frac{3}{10} = 0.3

Explanation:

The probability is the ratio of the number of blue marbles to the total number of marbles in the bag.

Problem 3:

Determine if the following data types are Discrete or Continuous:

  1. Number of apples in a basket.
  2. Time taken to run 100100 meters.

Solution:

  1. The number of apples is a countable whole number, so it is Discrete.
  2. Time can take any value within a range (e.g., 12.4512.45 seconds), so it is Continuous.

Explanation:

Discrete data typically involves counting, while continuous data involves measuring and can include decimals.