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

1. What is the average amount of time these students spent on homework?

2. Explain how you determined your answer.3. Does your answer reflect the mean, the

median, or the mode? Explain how you know.4. If you had found a different measure of central

tendency, would you expect your answer to be the same or different? Explain.

5. 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.