# Symmetry and Skew

Image Copyright 2013 by Passy’s World of Mathematics

This statistics lesson is all about Symmetry Skew and the Kangaroo !

The shape of a Histogram, Bar Chart, or Stem and Leaf plot tells us the type of data distribution we have.

If the tallest area (Mode) is in the middle of the Graph, with even reducing on each side of this, the Graph is called Symmetrical.

If the shape of the graph is not like this, and is bunched up on either the left or right side, we say the data is “Skewed”.

Skewed Data results for situations where there are either a lot of measured items that are very small, or a lot of items that are very large.

Skewed Graphs with their uneven shape affect the Mean Average Value.

This often means that we should use the Median as our average in these situations, and not use the distorted Mean value.

Positive Skew

The following example demonstrates and shows “Positive Skew”, where a Histogram stretches out to the right.

This particular class of students has not performed well on the Algebra Test, and so we have many low scores with tall bars on the left.

Image Copyright 2013 by Passy’s World of Mathematics

We can think of the shape of the Graph as resembling an Australian Kangaroo, and the direction of the Roo’s tail tells us the direction of the skew.

We call this the “Skew and Kangaroo Rule”.

Image Copyright 2013 by Passy’s World of Mathematics

Having an uneven skew shape in the Graph also affects the Mean,Median, and Mode.

The Mode is to the left of the middle Median, because the tall bars are all on the left hand side of the graph.

Due to the long tail of high values stretching to the right, the Mean will be dragged to the right of the Median.

Image Copyright 2013 by Passy’s World of Mathematics

Negative Skew

The following example demonstrates and shows “Negative Skew”, where a Histogram stretches out to the left.

This particular class of students has performed very well on the Algebra Test, and we have many high scores with tall bars on the right.

Image Copyright 2013 by Passy’s World of Mathematics

Placing a Kangaroo onto the graph shows that the tail goes to the left.

This means that the skew is to the Left, which is the negative direction.

Image Copyright 2013 by Passy’s World of Mathematics

Having an uneven skew shape in the Graph also affects the Mean,Median, and Mode.

The Mode is to the right of the middle Median, because the tall bars are all on the right hand side of the graph.

Due to the long tail of low scores stretching to the left, the Mean will be dragged to the left of the Median.

Image Copyright 2013 by Passy’s World of Mathematics

Symmetrical Histogram

If our Histogram is a pyramid type balanced shape, with both sides of the Median pretty much the same shape, then we say it is “Symmetrical”.

Image Copyright 2013 by Passy’s World of Mathematics

The “Kangaroo Rule” fits onto a symmetrical graph as shown below.

Image Copyright 2013 by Passy’s World of Mathematics

This particular maths class is fairly normal, and the sudents have produced both low and high scores.

The Mean, Median, and Mode will all be close together in the middle.

Image Copyright 2013 by Passy’s World of Mathematics

Skewed Histograms Video

The following video uses Microsoft Excel to show examples of Skewed Histograms, and how their shape affects the Mean Average Value.

Stem and Leaf Plot Symmetry and Skew

Measurement of Skew can also be applied to Stem and Leaf Plots.

However, we need to view the S&L Plot sideways to see whether the Skew is to the left or the right.

Image Copyright 2013 by Passy’s World of Mathematics

Image Copyright 2013 by Passy’s World of Mathematics

Image Copyright 2013 by Passy’s World of Mathematics

Image Copyright 2013 by Passy’s World of Mathematics

Summary

The shape of a Histogram, Bar Chart, or Stem and Leaf plot tells us the type of data distribution we have.

There is a great one page summary about Skew on the following web page:

http://www.mathsisfun.com/data/skewness.html

If the tallest area (Mode) is in the middle of the Graph, with even reducing on each side of this, the Graph is called Symmetrical.

If the shape of the graph is not like this, and is bunched up on wither the left or right side, we say the data is “Skewed”.

Skewed Data results for situations where there are either a lot of measured items that are very small, or a lot of items that are very large.

Skewed Graphs with their uneven shape affect the Mean Average Value.

This often means that we should use the Median as our average in these situations, and not use the distorted Mean value.

Kangaroo Statistics

Image Source: http://www.wildlifepark.com.au

The Australian Red Kangaroo is the world’s largest marsupial.

Females have one baby at a time, which at birth is smaller than a cherry.

Males are up to 1.8m (6 ft) tall and weigh up to 84 kg (187lbs).

Kangaroos continue to grow during all their life, which can be up to 20 years.

Giant short faced kangaroos (Procoptodon), extinct 40,000 years ago, were 3 m (10 feet tall) and weighed 200 kg.

Kangaroos live in groups made of 8 to 25 individuals, called mobs. A mob is made up of a male (the leader), several females and the young.

The stomach represents 15 % of the kangaroo’s weight.

Kangaroos can exist without water for up to 2 to 3 months.

Kangaroo’s back feet are made of two joined toes. The toe is extremely long bearing a large nail that makes a kangaroo’s kick extremely dangerous for dogs and even humans.

The tail is up to 1.1 m or 3.6 ft long in large kangaroos, and is used for leaning and for balancing during jumps.

Normal jumps are 1.2-1.9 m (4-6 ft) long, but at high speed they can be 13.5 m (45 ft) long and 3.3 m (11 ft) high.

The kangaroo can speed up to 50 km (30 mi) per hour, but it cannot sustain this speed for long periods.

Kangaroo Meat is one of the highest red meat protein sources and is very lean and healthy to eat.

There are set numbers for how many Kangaroos can be “harvested” yearly so that the natural population of Roos stays in environmental balance and is preserved.

“Harvested” means killed commercially for the meat and fur markets.

There are lots of statistics on Kangaroo Populations at the following Government Website:

For those worried about harvesting making Kangaroos in danger of extinction, statistics show that in 2010 there were around 25 million Kangaroos in Australia. Eg. More Kangaroos than People!

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