Measure of Central TendencyMeasures of central tendency – used to

organize and summarize data so that you can understand a set of data. There

are three common measures:Mean

MedianMode

Mean

Mean = sum of the data itemstotal number of data items

The symbol for the mean is Use the mean to describe the middle set of

data that DOES NOT have an outlier.Outlier – a data value that is much higher

or much lower than the other data values in the set.

Often referred to as the average.

MedianMedian – the middle value in the set when the

numbers are arranged in order.

If a set contains an even number of data items, the median is the mean of the two middle values.

Use to describe the middle of a set of data that DOES have an outlier.

Mode

Mode – the data item that occurs the most times.

Possible for a set of data to have no mode, one mode, or more than one mode.

Use the mode when the data are nonnumeric or when choosing the most popular item.

Range

• To calculate the range subtract the smallest data value from the largest data value.

• Example: 21, 15, 16, 25, 13, 18Range = 21 – 13 = 8

Examples• Find the mean, median, and mode. Which measure of central

tendency best describes the data.

Weights of textbooks in ounces:12, 10, 9, 15, 16, 10Mean = 12 + 10 + 9 + 15 + 16 + 10 = 72 = 12

6 6Median: 9, 10, 10, 12, 15 16 = 10 + 12 = 22 = 11

2 2Mode: 9, 10, 10, 12, 15 16 = 10

Since there is no outlier, the mean best describe the data

Range = 16 – 9 = 7

Mean Absolute Deviation (M A D) This measure averages the

absolute values of the errors.

List your data set (x) Find the mean of your data ( )

Find (x – ) Find |(x – )|

Find the sum of ∑|(x – )|.

Divide your sum by the total of your sample size (n)…

The mean absolute deviation is….

Math I Unit 4 Calculating the Mean Absolute Deviation (MAD)

Box-and-whisker plot (Boxplot)• A box-and-whisker plot is a visual way of

showing median values for a set of data.• The lower quartile (Q1) represents one quarter

of the data from the left and the upper quartile represents three quarters (Q3) of the data from the left.

• The five important numbers in a box-and whiskers plot are the minimum and maximum values, the lower and upper quartiles, and

the median.• The interquartile range = Q3 – Q1

How to construct Boxplot

Steps:1) Make a number line using equal interval

(from the minimum to the maximum value)2) Calculate the median Q23) Calculate Q1 = median of the first quarter4) Calculate Q3 = median of the third quarter5) Make a box connecting the quartile6) Draw a line from the minimum value to Q1

and another from Q3 to the maximum value

Example 4,6,8,10,12Minimum value = 4Median = 8Q1 = (4 + 6) ÷ 2 = 5Q3 = (10 + 12) ÷ 2 = 11Maximum value = 12Interquartile range = 11 – 5 = 6

Median( )

4 6 8 10 12

Xlargest

Xsmallest 1Q 3Q

2Q

Shape of a Distribution• Describes how data is distributed• Measures of shape

– Symmetric or skewed– See how mean compares to median, and possibly the

mode

Mean = Median(=Mode?) Mean < Median(< Mode?) (Mode<?) Median < Mean

Right-SkewedLeft-Skewed Symmetric

Distribution Shape and Box-and-Whisker Plot

Right-Skewed(positive)Mean>median

Left-Skewed(negative)Mean < median

Symmetric(zero)

1Q 1Q 1Q2Q 2Q 2Q3Q 3Q3Q