CS110 – Personal Computing 1

FORMULAS

CS110Personal Computing

Understanding formulas

The trainings are starting to introduce basic formulas and functions

We’re also starting to look at the statistical analysis part of the class

We noticed about ½ the class struggled with these basic formulas

So…

Percentages

If a city has 20% growth per year:

How do we calculate the estimated population?

By multiplying 20% times the current number

And add that to the current number

2013 has 4000 people

For 2014, that number is equal to :

Calculate the increase (4000 * 20%) = 800

Add that to 4000 = 4800

CS110 – Personal Computing 2

Let’s look at this

Why do we use absolute ref?

Let’s look at the action of copy and paste

When a relative address is copied and pasted

The references are updated

This allows the same formula to be used

When an absolute address is copied and pasted

The references are not updated

This allows constants or parameters to be used

Back to our example

CS110 – Personal Computing 3

Copying formula

Pasting formula

See how the relative references were updated

Note the absolute ref

It stays the same:

CS110 – Personal Computing 4

Ratios

What is a ratio?

a ratio is a relationship between two numbers of the same kind

Usually expressed A to B or A:B

What does it mean?

The ratio of bananas to apples is 1:4

The ratio of children to couples is 2.3:1

How do we compute ratios?

If we want to know the ratio of cars to people

We know there are 400 cars

We know there are 200 people

What is the ratio?

400:200

Can this be simplified?

Yes, by dividing the first by the second

Gives us 2:1

What do we use ratios for?

To compare two quantities

To simplify them to understandable numbers

The ratio of children to couples is :

57,500:25,000 or

2.3:1

Which is easier to understand

To extrapolate future numbers

If the ratio of bananas to apples in a delivery is 4:1

A delivery has 50 apples

CS110 – Personal Computing 5

Averages

How do we determine averages?

The sum of the values divided by the number of the values

So if :

Bob as 4 bananas

Terri has 2 bananas

Sammy has 3 bananas

What is the average number of bananas each of them has?

What do we use averages for?

To extrapolate numbers

If each person need an average of 2 liters /day, how much water will 10,000 people need?

The more samples there are, the more reliable the average

Bob – 5 liters

Sammy – 1 liters

Average is 3 liters

Small numbers have trouble with outliers

What is an outlier?

An observation that is numerically distant from the rest of the data

What causes outliers?

Error in measurement, outside influences

Example – if my study is tracking time to compute an algorithm, an outlier could be caused by someone else on the computer using the computer cycles or memory

CS110 – Personal Computing 6

How do they affect numbers?

They skew the average We record daily water

usage With Bob

Bob – 11 liters (watered his garden as well)

Sammy – 1.8 liters Tommy – 2 liters Andrea – 2.2 liters Average is 4.25 liters.

If we used this number for 10,000 people, we would have 22,500 liters too many

Without Bob Sammy – 1.8 liters Tommy – 2 liters Andrea – 2.2 liters Average is 2 liters

If we use this number, we would be right on.

How do we get rid of outliers?

First we have to identify them

We use standard deviation for this

Take those values out of the study

Re-compute the averages

Standard deviation

Used to identify the variance in the data

The formula for a complete population:

If N = population size and Avg = 1𝑁 𝑣𝑖𝑁

, then the standard

deviation is:

𝑖=1𝑁 𝑣𝑖 − 𝐴

2

𝑁

CS110 – Personal Computing 7

How does this apply to outliers?

We use the standard deviation to identify the outliers.

Let’s say we identify anything that is more than 2 times the standard deviation as an outlier

Let’s go back to our water example:

Water example

First we compute the average:

Bob – 11 liters (watered his garden as well)

Sammy – 1.8 liters

Tommy – 2 liters

Andrea – 2.2 liters

Jeff – 1.9 liters

Jackie – 2.1 liters

Average is 3.5 liters.

Water example

Next, we compute the sum of the squares

So Standard Deviation = 50.64 4

Compare to avg Deviation2

Bob 3.5 - 11 = -7.5 (-7.5)2 = 56.25

Sammy 3.5 – 1.8 = 1.7 (1.7)2=2.89

Tommy 3.5 – 2 = 1.5 (1.5)2 = 2.25

Andrea 3.5 – 2.2 = 1.3 (1.3)2 = 1.69

Jeff 3.5 – 1.9 = 1.6 (1.6)2 = 2.56

Jackie 3.5 – 2.1 – 1.4 (1.4)2 = 1.96

Total 67.6

CS110 – Personal Computing 8

Finishing standard deviation

With the sum of the squares – 67.6

67.6

6= 3.35

So our standard deviation is 3.35

Identifying the outliers

Standard deviation

3.35

2x standard deviation

6.7

Average = 3.5

Throw out anything

Over 10.2

Under -3.2

Value

Bob 11

Sammy 1.8

Tommy 2

Andrea 2.2

Jeff 1.9

Jackie 2.1

Value with Bob isgreater than 10.2

Re-compute the average

Value

Sammy 1.8

Tommy 2

Andrea 2.2

Jeff 1.9

Jackie 2.1

CS110 – Personal Computing 9

How to do this in Excel?

First, set up standard deviation

And average

Eliminate the outliers

CS110 – Personal Computing 10

And now compute new average

Confidence interval

Given in a range and a percentage.

Says that with the percentage given, the actual value will fall within the range

I.E. given:

A normal distribution

a 95% confidence interval

A range between 9 and 11

Says that at least 95% of the time, the value calculated will fall between 9 and 11.

What is a normal distribution?

CS110 – Personal Computing 11

How is this helpful?

When you calculate averages, you assign them a confidence interval and a range.

This tells your readers how reliable your data is

Calculated by:

Assumes normal distribution

Standard deviation

Given a specific confidence level

Number of data point

Calculate in Excel

Improve the interval

By increasing the number of samples

CS110 – Personal Computing 12

Improve the intervalDecrease the confidence level

Median

The numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half.

The median can be used as a measure of location when a distribution is skewed

when end-values are not known

when one requires reduced importance to be attached to outliers

A disadvantage of the median is the difficulty of handling it theoretically

Median

Given the following sequence, what would be the median of the values:

1,2,2,3,3,3,4,5,5,6,6,6,14

CS110 – Personal Computing 13

In Excel

Mode

In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution

Very useful for discreet functions rather than continuous functions

If more than one, the series can described as bimodal or multimodal

Mode

What would be the mode of the series:

1,2,3,3,4,5,6,7,7,7,8,8,9,10,10,11,12

How about:

1,2,3,3,4,5,5,6,6

1,2,3,3,4,5,5,5,6,6,6

CS110 – Personal Computing 14

In Excel

References

1. http://en.wikipedia.org/wiki/Ratio

2. http://en.wikipedia.org/wiki/Standard_deviation

3. http://en.wikipedia.org/wiki/Confidence_interval

4. http://en.wikipedia.org/wiki/Median

5. http://en.wikipedia.org/wiki/Mode