Quartiles And Box Plots Common Core Algebra 1 Homework Help

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Often times, we need to know more about our data than just a possible center value.
One of the additional pieces of information that we may need is the actual distribution of the data (how the data is spread out). To find this information, we examine the data for a
five statistical summary (or five number summary): (1) minimum, (2) maximum, (3) median (second quartile), (4) first quartile, and (5) third quartile. These pieces of information will show the extent to which the data is located near the center or near the extremes of the set.

Before we look further at a five statistical summary, let's refresh our skills on quartiles.



A median divides a data set into two equal parts. The set can be subdivide further into four equal parts, by values called quartiles. The quartiles divide the data set into quarters, with each quarter containing one-fourth (or 25%) of the data. The quartiles are like additional "medians" of the lower and upper halfs of the data set. A quartile is a number, it is not a range of values. Data can be described as being "above" or "below" the first quartile, but data is never "in" the first quartile.

Q1: The first quartile is the middle (the median) of the lower half of the data set. One-fourth (25%) of the data lies below the first quartile, and three-fourths (75%) lies above.

Q2:The second quartile is another name for the median of the entire set. One-half (50%) of the data lies below the second quartile, and one-half (50%) lies above.

Q3:The third quartile is the middle (the median) of the upper half of the data set. Three-fourths (75%) of the data lies below the third quartile and one-fourth (25%) lies above.

The difference between the third quartile and first quartile is called
the interquartile range (IQR).
The interquartile range (also called the midspread or middle fifty), is the distance between the third and first quartiles and is considered a more stable statistic than the "range" of the set. 
The IQR contains 50% of the data. 

For the example shown above, the IQR = 51 - 26½ = 24½.

It may be the case that a data value falls well outside the range of the other values in the set. Such data values are called outliers (as they "lie outside" the other values) . We will see, later on this page, that outliers may lead to false impressions regarding the distribution of a data set.

Outliers are defined as those data points that fall more than a specified distance from the first or third quartiles. That specified distance is 1.5 • IQR (one and one-half times the IQR). Data points that fall to the far left, or far right, of an ordered data set should be tested as possible outliers.

Outliers are:
greater than Q3 + (1.5 • IQR)
(referred to as the upper fence)
or less than Q1 - (1.5 • IQR)
(referred to as the lower fence)

Five Statistical Summary


Let's describe our data set (discussed above) with a five statistical summary:
minimum, maximum, median, first quartile and third quartile.
DATA SET: {24, 25, 26, 27, 30, 32, 40, 44, 50, 52, 55, 57}

While not telling every value in the data set,
a five statistical summary will tell you that:
• half (50%) of the data values are below 36,
• half (50%) of the data values are above 36, and
• half (50%) of the scores are between 26½ and 51.
It also tells how the data break out in quarters,
along with the smallest and largest data values.

For calculator help with
five-number summary
click here.


Box & Whiskers

A five statistical summary can be represented graphically as a box and whisker plot (or box plot). The first and third quartiles are the ends of the box, the median is indicated with a vertical line in the interior of the box, and the minimum and maximum are the ends of the whiskers (unless an outlier is present). Each of the four "sections" of a box plot represents 25% of the data in the set.

How to construct a box and whisker plot by hand:

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