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### Dealing with Data

7th grade math

What is data?

- Data is information.
- Raw data can come in many different forms, the two most common are:
- Categorical data – data with specific labels or names for categories (usually in word form)
- Numerical data – data that are counts or measures (usually in number form)

Variability

- Variability – indicates how widely spread or closely clustered data values are
- Students collect data on the amount of change in the pocket of every student at NHM. (Clustered or spread?)
- Students survey current students at NHM to find out their grade level – 6th,7th, or 8th.

(Clustered or spread?)

How do you display data?

- The easiest way to display data is in a graph or chart.
- Pictograph Circle Graph
- Histogram Line Plot
- Bar Graph Scatter Plot
- Line Graph Box-and-Whisker Plot
- Frequency Distribution
- Stem and Leaf Plot

What makes a good graph?

- A good graph…
- Fits the data you have collected.
- Has a title and labels.
- Accurately displays your data.
- Allows a reader to easily draw conclusions.
- Catches the reader’s attention.
- Is easy to read and understand.

Where does data come from?

- ????
- Surveys
- Studies
- Questionnaires
- Census data

Populations, Samples, and Statistics

- Population – the entire set of items from which data can be selected (ex. Every 7th grade student, every girl at NHM)
- If we collected data from EVERY member of a population we would refer to this as a census.
- Collecting data from an entire population can be a long and difficult process, but the data obtained would be extremely accurate and reliable.

Populations, Samples, and Statistics

- Sample – a selected group of a population that is representative of the entire population. (ex. Twenty 7th grade students in Mr. Ridley’s math class)
- Samples can be:
- Random – data is obtained from random members of a population
- Systematic – data is obtained using a system for selection (ex. Every 10th person)
- Convenient – data is obtained from the easiest source available within your population (ex. People who sit next to you in class)

Populations, Samples, and Statistics

- Anytime you obtain data about a measured characteristic of your sample, you have collected a statistic.
- If you obtain data about a measured characteristic of an entire population, you have collected a parameter.
- If you find a data point that is not consistent with your other results (way too high, way too low) we call it an outlier and it can be removed.
- Which data would be more reliable?

Interpreting Data

- Raw data does not come in a user-friendly format.
- It must be processed and presented in a form that is easy to read and understand.
- One system for doing this is graphing, which allows for a visual picture of a data set.

Measures of Central Tendency

- Another system for interpreting data are the measures of central tendency.
- Also called measures of center, these numbers attempt to summarize a data set by describing the overall clustering of data in a set
- The goal of these numbers is to find one single numerical value that can represent the “average” value found in the entire set.

Measures of Central Tendency

- The 3 most common measures are:
- Mean – the average, found by dividing the sum of all the numbers in a data set by the number of pieces of data you collected.
- Median – the middle value, found by locating the middle number in a ordered data set
- Mode – the most common value, found by locating the most frequently appearing value in a data set

Tricks of the Trade

- Median – the cross out method
- Order your data set from least to greatest
- Repeatedly cross out the smallest and largest value in your data set until you arrive at the median
- If you have two values left, add them together and divide by two.
- Mode – it’s the “MOST”
- Both four letter words
- Both begin with MO

Tricks of the Trade

- Mean – sorry =(
- I really am sorry, but you just have to do the math.
- Add them up, divide by the number of pieces of data in your set.

Practice

- Its almost report card time and Sam is worried about his grade. He has made the following scores on his 7 tests in math: 77, 84, 83, 78, 92, 90, 84. Help Sam out by finding his …
- Mean
- Median
- Mode

Practice

- Sam’s football coach told him he was going to be benched if his grade was below a “B”, should Sam be worried? Explain.
- Which measure of central tendency would give Sam the best grade possible?
- Which measure of central tendency best reflects Sam’s actual test performance?
- Are there any outliers in his test scores?

Practice – On your own

- A statistician randomly selected 12 7th grade students and asked them how much time they spend each night on homework. The responses were:
- 0 mins 20 mins 15 mins
- 1 hour 30 mins 45 mins
- 15 mins 0 mins 15 mins
- 30 mins 1 hour 1 hr & 10 mins

Practice – On your own

- What is the average amount of time these students spent on homework?
- Explain how you determined your answer.
- Does your answer reflect the mean, the median, or the mode? Explain how you know.
- If you had found a different measure of central tendency, would you expect your answer to be the same or different? Explain.
- If a 7th grader spends 15 hours per day at home, what percent of home time does the “average” student spend on homework?

Measures of Variability

- Attempt to describe the clustering seen in a set of numbers.
- The two most common measures of variability are:
- Range (easy)
- Interquartile Range (complicated)
- Range is used quite often, interquartile range is really only seen when creating a box-and-whisker plot

Range

- Range is quite simply the difference between the largest value and smallest value in a numerical data set.
- Code word: difference = subtraction
- EX. 12, 15, 19, 21, 41, 67
- The range is the largest value (67) minus the smallest value (12), which equals 55.

Interquartile Range

- Yes, it is as complicated as it sounds.
- First, what is a quartile?
- Think quad, which means four.
- Ok, so 4 of what?
- Quartile refers to one of 3 numbers that can break a set of data into 4 even sections.
- Quartile – a number that creates 4 equal sections of numbers in a distribution

Interquartile Range

- Lets see these quartiles in action!
- Step 1: Put a set of numbers in order
- 13, 15, 16, 18, 22, 25, 26
- Step 2: Find the median
- 13, 15, 16, 18, 22, 25, 26
- This separates the data into two sections, exclude the median
- [13, 15, 16] 18 [22, 25, 26]
- The median is now called the Second Quartile or Q2.

Interquartile Range

- Step 3: Find the median of the set of numbers less than Q2.
- [13, 15, 16] 18, 22, 25, 26
- 13, 15, 16
- This number is now called the First Quartile or Q1.
- Step 4: Find the median of the set of numbers greater than Q2.
- 13, 15, 16, 18, [22, 25, 26]
- 22, 25, 26
- This number is now called the Third Quartile or Q3.

Interquartile Range

- Step 5: Find the distance between the Third Quartile and the First Quartile
- (Q3 – Q1)
- 13, 15, 16, 18, 22, 25, 26

Q1 Q2 Q3

(25 – 15) = 10

This value is the interquartile range!

Interquartile Range

- So why did we do all of that work?
- What does a range tell us?
- All values fall between the smallest and largest value……..well duh!!!
- What does the interquartile range tell us?
- Half (50%) of all values fall between the first and third quartile.
- The interquartile range reflects the real “heart” of the data set.

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