Good morning! Please get out your homework for a check. Be prepared to share your group’s work. Warm Up: Do #40 (handout on desk). 

1.2 Describing Distributions Numerically

Ways to display data – Cumulative Frequency Graph Decide on class intervals and determine frequencies. To determine cumulative frequencies, add the class frequency to the prior class frequencies to get a total.

Ways to display data – Ogive (Relative Cumulative Frequency Graph) Set up a table to calculate for each interval: Frequency (# times occurs) Relative Freq. (Freq. Divided By Total #) Cumulative Relative Freq.

Histogram to Ogive 61-70 71-80 81-90 91-100 Class Freq Cumulative Frequency Cumulative Rel. Freq 61-70 71-80 81-90 91-100

Let’s look at #20 a and b

We last looked at different displays we can use to represent data We last looked at different displays we can use to represent data. Today, we’ll begin looking at how to describe data. Because we are describing data, we call this branch of statistics “descriptive statistics.” In statistics, we’re often concerned with two things - where is most of the action happening and why isn’t ALL the action happening there. The first deals with the concept of “Central Tendency” and the second deals with the concept of “Deviation” or “spread.”

Measures of Central Tendency Mean Median Mode

Mean Arithmetic average The mean always exists The mean does not have to be one of the data values The mean uses all the data values The mean is affected by extreme values. (Not resistant to outliers.)

Median M; Midpoint of the data after it’s been sorted. If even # of data points, average the middle two. The median always exists. The median does not have to be one of the data values. The median does not use all of the data values, only the one(s) in the middle. The median is resistant to change, it is not affected by extreme values.

Mode The mode is the most frequent value. If no value appears more than any other, then there is no mode. If two or more values appear more than the others, then the data is bimodal or multimodal. The mode may or may not exist. If it does exist, there may be one or several modes. The mode has to be one of the data values. The mode does not use all the data values. The mode is probably not affected by extreme values since it is unlikely the extreme values are not the most common.

Resistance: how well a statistic remains unaffected by outliers

Mean Example: Find the mean of 1, 3, 5, 2, 2, 10,000 Solution: 1668.83 Question: Does this number seem fairly representative of the data? Answer: No. Most of the data is clustered between 1 and 5 with a single outlier at 10,000 but our mean was 1668.83, which isn’t close to any of our data. Advantages: Disadvantages: Easy to find Can be thrown off by Gives quick idea of skewed data (i.e. outliers) the expected value

How do we know where the mean, median, and mode are on a graph?

Mean = Median = Mode Symmetric

Mean < Median < Mode Skewed Left

Mean > Median > Mode Skewed Right

Centers and Spread Mean equal balance point Is used with symmetric data Median equal areas point Is used with skewed data Range High value - Low value Advantage: Quick sense of the data Disadvantage: Can be influenced by outliers (not resistant)

Measures of Spread Variance Meaning: Average of the squares of the deviations from the mean Standard deviation s (the square root of the variance) Meaning: Average distance from the mean Example: Suppose the mean of a data set was 20 with a standard deviation of 5. This means that on average, a data point falls 5 units away from the mean.

Let’s Calculate Age Freq 5 1 6 2 7 Mean x-bar = 6 Variance s2=0 .667 Standard Deviation s = 0.8165

Assignment HW Read pp 54-73 Do Exercises #’s 44, 45, and for # 47 Compare and Contrast the calorie distributions of Meat and Poultry Hotdogs. (Box Plots indicates Modified from now on) Review #24,33, 35, 43, 55, 61, 66, 73