13.2: Measuring the Center and

Variation of Data

Kalene Mitchell

Allie Wardrop

Sam Warren

Monica Williams

Alexis Carroll

Brittani Shearer

Common Core Standards

Summarize and describe distributions. Giving quantitative measures of center

(median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

CCSS.MATH.CONTENT.6.SP.B.5.C

Grade 6

The Mean…

Definition: The mean or average, of a collection of values is x8 = S/n, where S is the sum of the values and n is the number of values. The symbol x8 should read as “x bar.”

Commonly referred to as: Average

Arithmetic mean

The Mean…

Example: Find the mean of the following values:

13, 18, 13, 14, 13, 16, 14, 21, 13

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All 12 players on the Uni Hi basketball team played in their 78- to-65 win over Lincoln. Jon Highpockets, Uni Hi’s best player, scored 23 points in the game. How many points did each of the other players average?

The Median… Let a collection of n data values be written in

order of increasing or decreasing size. Largest # Smallest #

Half values above & half values below

Can be used in situations where we cannot get a proper measurement but can only rank data in order ex: Arrange workers in order of their performance.

A worker in the middle would represent median performance.

50% of the workers do not work as hard

50% of the workers work harder

The Median…

If n is odd, the median, denoted by x, is the middle value in the list.

X= middle Value

If n is even, x is the average of the two middle values. The symbol x should be read as “x hat.” X= n1 (first middle #) + n2 (second middle # /

2)

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The Median…

Example: Find the median of the following values:

11, 13, 5, 19, 33, 12, 14, 15, 16, 11, 10, 32

If we take away one values, how would it change the problem?

11, 13, 5, 19, 33, 12, 14, 15, 16, 11, 10

The Mode…

Definition: A value that occurs most frequently in a collection of values.

If two values occur equally often and more frequently than all other values, there are two or more modes. (Bi….Tri…)

Commonly referred to as: The number seen most often

The Mode..

Example: Find the mode of the following values:

44, 35, 36, 43, 41, 40, 35, 37, 34, 36, 37, 33, 36

Measures of Variability

Variability- how far spread out the scores or data points are

There are four frequently used measures of variability

Range: highest minus lowest score

Interquartile range- range of middle 50% scores

Standard deviation

Variance

Upper and Lower Quartiles

Definition: Consider a set of data arranged in order of increasing size. Let the number of data values, n, be written as n=2r when n is even, or n=2r+1 when n is odd, for some integer r.

The lower quartile, denoted by QL, is the median of the first r data values.

The upper quartile, denoted by Qu, is the median of the last r data values.

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For the given data set of this problem, determine the lower and upper quartile.

A = (12,7,14,15,9,11,10,11,0,8,17,5)

Interquartile RangeIQR

Definition: Difference between the upper and lower quartile

IQR= Qu - QL

In other words: 75th percentile – 25th percentile

Outlier

Definition: data value that is LESS THAN QL –(1.5 x IQR) or GREATER THAN +(1.5 x IQR)

In other words, data points or scores that are atypical of the other values in the data set.

So less than 25th percentile – (1.5 x 75th percentile)

Or greater than 75th percentile – (1.5 x 25th percentile)

Box Plot or Box-and-Whisker Plot

Definition: consists of a central box extending from the lower to the upper quartile, with a line marking the median and with line segments, or whisker, extending outward from the box to the extremes.

Standard Deviation

Definition: Let X1, X2, X3…..Xn be the values in a set of data and let x denote their mean. What it means- How far from the normal

Formula:

How to Find Standard Deviation

Step 1: Find the mean

Step 2: Find the difference each number is from the mean

Step 3:Take each difference and square it

Step 4: Add those numbers together

Step 5: Divide that sum by the total number of terms

Step 6:Take the square root of that number