Section 12.2: Statistics and Parameters. You analyzed data collection techniques. Identify sample statistics and population parameters. Analyze data sets.

Chapter 12: Statistics and Probability

Section 12.2: Statistics and Parameters

Then/Now

You analyzed data collection techniques.

• Identify sample statistics and population parameters.

• Analyze data sets using statistics.

Statistical inference- when the statistics of a sample are used to draw conclusion about the entire population

Statistic- a measure that describes a characteristic of a sample

Parameter- a measure that describes a characteristic of a population (typically estimated based on the statistics of a carefully chosen random sample)

Vocab:

Statistics can and usually change from sample to sample.◦ Ex: The number of girls to boys in any given

classroom at McKinley Parameters will not change.

◦ Ex: The number of girls to boys in McKinley

Notes:

Example 1Statistics and Parameters

A. Identify the sample and the population for each situation. Then describe the sample statistic and the population parameter.

A movie rental business selects a random sample of 50 orders in one day. The mean number of rentals per order is calculated.

Answer: sample: 50 movie orders; population: all movie orders for the day of the sample; sample statistic: mean number of rentals per order in the sample; population parameter: mean number of rentals per order for all rentals the day of the sample

Example 1Statistics and Parameters

B. Identify the sample and the population for each situation. Then describe the sample statistic and the population parameter.

A stratified random sample of 2 trees of each species is selected from all trees at a nursery. The mean height of trees in the sample is calculated.

Answer: sample: 2 trees of each species found at the nursery; population: all trees at the nursery; sample statistic: mean height of trees in the sample; population parameter: mean height of all trees at the nursery

Means, Medians, and Modes are measures of central tendency that represent data.

Measures of variation that assess the variability of the data can also represent the data.◦ Examples: Range, Quartiles, Interquartile Range,

Mean Absolute Deviation, and Standard Deviation

Notes

(MAD) is the average of the absolute values of the differences between the mean and each value in the data set.

The ‘MAD’ is used to predict errors and judge how well the mean represents the data.

Mean Absolute Deviation

Key Concept

Example 2Mean Absolute Deviation

PETS A rescue agency records the number of pets adopted each month: {14, 18, 12, 17, 15, 20}. Find and interpret the mean absolute deviation.

Step 1Find the mean.

Step 2Find the absolute values of the differences.

Example 2Mean Absolute Deviation

Step 2Find the absolute values of the differences.

Example 2Mean Absolute Deviation

Step 3Find the sum.

2 + 2 + 4 + 1 + 1 + 4 = 14

Step 4Find the mean absolute deviation.

Formula for Mean Absolute Deviation

The sum is 14 and n = 6.

Example 2Mean Absolute Deviation

Answer: A mean absolute deviation of 2.3 indicates that, on average, the monthly number of pets adopted each month is about 2.3 pets from the mean of 16 pets.

In a set of data, the standard deviation shows how the data deviate from the mean. A low standard deviation indicates that the data tend to be very close to the mean, while a high standard deviation indicates that the data are spread out over a larger range of values.

The standard deviation is represented by the lowercase Greek letter sigma, σ.

The variance of the data is the square of the standard deviation

Standard Deviation

Key Concept

Example 3Variance and Standard Deviation

SCORES Leo tracked his homework scores for the past week: {100, 0, 100, 50, 0}. Find and interpret the standard deviation of the data set.

Step 1Find the mean.

Example 3Variance and Standard Deviation

()2 = 2500

(50 – 0)2 = 2500

(50 – 100)2 = 2500

(50 – 50)2 = 0

(50 – 0)2 = 2500

Step 2Find the square of the differences, .

Step 3Find the sum.

2500 + 2500 + 2500 + 0 + 2500 = 10,000

Example 3

Step 4Find the variance.

Variance and Standard Deviation

Formula for Variance

The sum is 10,000 and n = 5.

Example 3

Step 5Find the standard deviation.

Answer: A standard deviation very close to the mean suggests that the data deviate quite a bit. Most of Leo’s scores are far away from the mean of 50.

Variance and Standard Deviation

Square Root of the Variance