UNIT 8: Statistical Measures Statistics practice of analyzing a set of data Measures of Central Tendency: numbers that represent the middle of the data Mean ( x ): Arithmetic average (Total Data of Values over Number of values) Median: Middle of the data listed in ascending order (Count inward from Max and Min – if not an exact data value, find midpoint of last two data values) Mode: Most commonly occurring number(s) (MORE THAN ONE value if repeated same number of times or NONE if no repeated values)

Measures of Variation: Variance, Standard Deviation Dispersion: How spread / scattered a set of data is Range: Difference between the highest and lowest data value (MAX – MIN) Inner Quartile Range (IQR): The difference between Q3 and Q1 (Q3 – Q1) Outlier: Any data item outside Q1 or Q3 a distance 1.5*IQR *Standard Deviation (σ): A measure of how much the data is spread out **We don’t use variance** Variance (σ2 ): Measures how much the data differs from the mean

5 Number Summary: Min: Q1: Med (Q2): Q3: Max: Minimum Value (0 Percentile) Q1: Quartile 1 (25th Percentile) Med (Q2): Median (50th Percentile) Q3: Quartile 3 (75th Percentile) Max: Maximum or Q4 (100th Percentile) Min Med Max Q1 Q2 Q3 Q4

Calculator Commands: One-Variable Statistics Input data: [STAT]  [EDIT]  L1 [STAT] → [CALC] → [1: 1-Var STATS] σx → represents standard deviation = represents mean n = number of entries med = median Etc…

Ex: Find the mean, median, mode, and standard deviation of … (2) GPAs: 3.42, 3.91, 3.33, 3.57, 3.45, 4.0, 3.65, 3.71, 3.35, 3.82, 3.67, 3.88, 3.76, 3.41, 3.62 (1)Test Scores: 85, 76, 88, 91, 85, 58, 88, 91, 97, 91, 88, 97, 97 Mean = 87.0769308 Median = 88 Mode = 97 (x3) Stand. Deviation = 10.08015215 Mean = 3.6366666 Median = 3.65 Mode = None Stand. Deviation = 0.2058370445

Normal Distribution Mean = Median “Bell Curve” Mean + Median Skewed data is described based on location of tail Positively Skewed (Right) “Mean is to Right of Median” Negatively Skewed (Left) “Mean is to Left of Median” Median Mean Mean Median

Biased vs Unbiased When you are sampling a section of the population (giving a poll), there are good and bad ways to do it. An unbiased sample is one in which a good section of the population is represented. Entire population is well represented A biased sample is one in which the sample does not adequately represent the population. A group or section of population is targeted or missing The bigger the sample, the more accurate the results (larger samples more closely reflect the population).

Determine if situation would produce (un)biased sample: Ex. 1: You want to determine how many people in a school are going to college, so you ask every third person in an AP Calculus class. Ex 2: You want to find out people’s favorite kind of food, so you ask 100 people at the food court at the mall. Ex 3: Putting the names of all the seniors in a hat, then drawing names from the hat to select a sample of seniors. Ex 4: Determining the shopping preferences of the students at your school by asking people at the mall. Ex 5: Finding the average height of the students in your school by using the members of the football team. BIASED UNBIASED UNBIASED BIASED BIASED

Math 3 Hon: Unit 8 Statistics The Margin of Sampling Error (ME) is a numerical way to determine the difference between how a sample and the population responds. Math 3 Hon: Unit 8 Statistics If p represents the percentage of people with a particular response from a sample of n people, then 95% of the time the population will respond within one ME of the response p (p – ME or p + ME)

Exp 1: 1500 people were asked a question and 38% responded “yes”. – the margin of error (ME) is 2.5% That means that there is a 95% chance that the people in the population that would answer “Yes” between (38-2.5) = 35.5% and (38 + 2.5)= 40.5% of the time. (38 % with a margin of error of 2.5%) Exp 2:. p = 16%, n = 400 Exp 3: 934 out of the 2150 students said they read the newspaper every day