Algebra 2 mean,median,mode

Concept of Median

• The median is the middle value in a sorted dataset.

• For an odd number of elements, the median is the middle number.

• For an even number of elements, the median is the average of the two middle numbers.

  • Example: Data set = [19, 20] → Median = (19 + 20) / 2 = 19.5

Concepts of Mean and Mode

• The mean is calculated by adding all values and dividing by the count of values.

  • Example: Mean of [19, 20] → (19 + 20) / 2

• The mode is the most frequently occurring number in a dataset.

  • Example: In [17, 18, 18, 20], 18 is the mode.

Outliers

• An outlier is a value that significantly differs from the rest of the dataset.

  • Example: In [16, 17, 18, 54], 54 is an outlier.

• There are no strict rules for defining outliers; it's typically based on intuition and context.

Choosing Between Mean and Median

• The median is often a better measure of central tendency in skewed datasets or those with outliers since it isn't affected by extreme values.

• Example: Outlier 54 skews the mean, thus making median a better representation of central value, especially in age datasets.

Minimum and Maximum

• Minimum: The smallest value in the dataset.

  • Example: In [16, 17, 18, 20], the minimum is 16.

• Maximum: The largest value in the dataset.

  • Example: In [16, 17, 18, 20], the maximum is 20.

Range

• The range indicates the spread of the dataset and is calculated as:

  • Range = Maximum - Minimum.

  • Example: Range of [16, 54] = 54 - 16 = 38.

Quartiles

• Quartiles divide the dataset into four equal parts:

  • Q1 (1st quartile): Median of the lower half of data.

  • Q2 (2nd quartile): The median of the dataset.

  • Q3 (3rd quartile): Median of the upper half of data.

• To find Q1 and Q3, the data should be in order:

  • Example Data: [16, 17, 18, 19, 20, 22, 23, 30, 54]

  • Q1 = median of [16, 17, 18, 19] → 18

  • Q3 = median of [20, 22, 23, 30, 54] → 22

Interquartile Range (IQR)

• The IQR is the difference between Q3 and Q1 and measures the spread of the middle 50% of the data:

  • IQR = Q3 - Q1.

  • Example: IQR = 22 - 18 = 4.

• Important for understanding data spread especially when there's an outlier present.

Percentiles

• Percentiles indicate the relative standing of a value in a dataset.

• Example: If a score is in the 80th percentile, it means the score is higher than 80% of the data points.

• Percentiles can be determined by sorting data and identifying percent divisions.

Measures of Central Tendency vs. Measures of Spread

• Measures of Central Tendency: Mean, Median, Mode (represents the center of a dataset)

• Measures of Spread: Range, IQR (describes how spread out the values are)

• It's important not to confuse the two when analyzing data.