Summary Statistics

Vocabulary

Center – the middle of your set of data; represented by , ̧ and/or .

Spread – the variability of your set of data; represented by , _______, _______, and .

Use the data below to calculate the statistics

9, 3, 2, 7, 12, 5, 6, 6, 8

Put in order from least to greatest

Minimum - the smallest number. Maximum - the largest number.

Mean - measures the center of the data. Calculated by adding up all of the numbers and dividing by the number of data.

Median - measures the center of the data. Put the numbers in order from least to greatest. Find the number that is in the center. If there are two numbers in the center, average them together.

Mode - measures the center of the data. The number that appears the most.

Lower Quartile and Upper Quartile - Use the median to split the data into two. Do not include the median in either side of data. Find the median of each half. The median of the lower half is the lower quartile (Q1). The median of the upper half is the upper quartile (Q3).

Range - The spread of the data. Subtract the minimum from the maximum.

Interquartile Range (IQR) - The spread of the middle 50% of the data. Subtract the lower quartile from the upper quartile.

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Standard Deviation

A measure of spread and variability of a data set. Used for normal distribution and outlier calculations.

Find the standard deviation of the data set below

{6, 8, 7, 5, 10, 6}

Based on a sample, we extrapolate the results of a statistics to try to estimate the parameters of the population. We do this by setting up a proportion.

= __

Example 1: A car factory just manufactured a load of 6,000 cars. The quality control team randomly chooses 60 cars and tests the air conditioners. They discovered that 2 of the air conditioners do not work. How many of the manufactured cars do you expect to have broken air conditioners?

Example 2: In a survey of 40 employees at a company, 18 said they were unhappy with their pay. The company has 180 employees. How many employees do you expect are unhappy with their pay?

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On your calculator:

Step 1: Under “STAT”, choose “Edit…” and put data into a list .

Step 2: Under “STAT”, choose “CALC” and then “1-Var Stats”.

Step 3: Choose the correct list, or type a comma then the list name.

Sampling Methods

Vocabulary

___________________ – a portion of the population.

___________________ – the entire group that is being studied.

___________________ – a number that describes a sample.

___________________ – a number that describes the population.

___________________ – a study of a sample of the population used to predict characteristics of the entire population.

___________________ – a study of the entire population.

______________________________ – a random sample that is representative of the entire population.

______________________________ – a sample where one or more groups of the population ARE NOT INCLUDED.

______________________________ – a sample where one or more groups has a greater representation than in the general population.

Name Description Example

Every member of the population has an equal chance of being chosen.

Pick sample participants based on a rule.

Choose participants based on who is available.

Participants choose to be included in the sample.

The population is divided into groups. Each member of each group has an equal chance of being chosen.

The population is divided into subgroups; then an entire subgroup is chosen at random to be the sample.

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Identify the Population and the Sample in each of the following.A car factory just manufactured a load of 6,000 cars. The quality control team randomly chooses 60 cars and tests the air conditioners. They discovered that 2 of the air conditioners do not work.

Population: Sample:

In a survey of 40 employees at a company, 18 said they were unhappy with their pay. The company has 180 employees.

Population: Sample:

Biased or Unbiased? Type of sampling method?Call 100 random people listed in the phone book to ask how long they’ve been living in their home.

Ask every 10th person at a school off of a complete roster about how many courses they are taking.

Ask 25 families eating at a restaurant on a Tuesday night about how often they eat out.

Out of each class at school, ask a number of students proportional to the number of students in each class how much they enjoyed homecoming.

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Normal Distribution

Vocabulary

Normal Distribution – one type of distribution of data, which describes how data clusters around the mean. It is and , with the peak at the .Examples: heights of people, blood pressure, IQ, body temperature, etc.

- A normal distribution comes from the distribution of a ___ , variable.

- In a normal distribution, the , , and all exist at the center of the curve.

The Empirical Rule applies to all Normal distributions. It dictates the following:

______% of data falls within 1 s.d. from the mean

______% of data falls within 2 s.d. from the mean

______% of data falls within 3 s.d. from the mean

You have a sample with x = 15 and σ = 2.2

Draw a normal distribution demonstrating the empirical rule.

Within what range will 68% of your data fall?

Within what range will 95% of your data fall?

Within what range will 99.7% of your data fall?

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A normal distribution has a mean of 27 and a standard deviation of 5. Find the probability that a randomly selected x-value from the distribution is in the interval of 17 to 37.

The distribution of heights of young women aged 18 to 24 is approximately normal with a mean (x) of 64.5 inches and a standard deviation (σ) of 2.5 inches.

A) Draw a Normal Distribution curve to represent this data, clearly showing the application of theempirical rule.

B) What % of women are taller than 69.5 inches?

C) Between what heights are the middle 95% of women?

D) What % of women are shorter than 62 inches?

E) A height of 67 inches corresponds to what percentile of adult female American heights?

Scores are normally distributed with x = 650 and σ = 100.

A) Draw the Normal Distribution curve that represents this data.

B) What is the probability that a randomly selected test score is between 450 and 850?

C) Out of 1000 randomly selected test scores, how many would you expect to be between 450 and 850?

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D) Out of 2300 randomly selected test scores, how many would you expect to be between 650 and 950?

Z-Scores

Vocabulary

Z-Score – the number of standard deviations away from the mean. We use z-scores to tabulate the area under the curve and evaluate percentiles.

A z-score is the number of standard deviations a value is from the mean.

The STANDARD NORMAL DISTRIBUTION is the specific normal distribution with a mean of 0 and a standard deviation of 1.

If our Normal Distribution has x-values with a mean of x and a standard deviation of σ, then these x-values can be converted into their corresponding “z-scores”. These z-scores allow us to find the probability of a certain range of values within the original Normal Distribution.

If a set of data is normally distributed with a mean of x and a standard deviation of σ, then the standard normal value (z-score) is calculated as

z =

These z-scores are standardized values that allow data with different units to be compared. (Ex. ACT to SAT score)

The z-score tells us how many the original value is from the mean.

Observations larger than the mean have z-scores and observations smaller than the mean have z-scores.

After calculating the z-score, it can be analyzed using a z-score table. The values in a z-score table give the to the of that z-score.

NEGATIVE: POSITIVE:

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Find the z-score of a data point of 21 when the mean is 24 and the standard deviation is 2.

What percent is to the left of 21? What percent is to the right of 21?

If a data point has a z-score of .2, what is the probability of getting lower than that score?

If a data point has a z-score of –.16, what is the probability of getting higher than that score?

What is the probability of getting a score between the z-score of –3.22 and the z-score of 1.9?

Scores on a test are normally distributed with a mean of 75 and a standard deviation of 8.

A) Estimate the probability that a randomly selected student scored less than an 87.

B) Estimate the probability that a randomly selected student scored higher than a 79.

C) Estimate the probability that a randomly selected student scored between 71 and 75.

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