Statistics Boot Camp: Student PacketStatistics Basics

Measures of central tendency:

Find the mean, median, and mode of the data, if possible. If any of these measures cannot be found or a measure does not represent the center of the data, explain why.

A sample of seven admission test scores for a professional school are listed below:

10.6 9.9 11.2 9.9 10.3 9.9 10.9

What is the mean score?

The mean score is _____

Does the mean represent the center of the data? Answer: Yes or No. ______

What is the median score? _______

Does the median represent the center of the data? Answer: Yes or No: ______

What is the mode of the scores? ______

Does the mode represent the center of the data? Answer: Yes or No ______

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Remember:

Mean = µ “Population mean” = ∑ of all values∈data setnumber of values∈dataset

Mean = x“Sample mean” = ∑ of all values∈data setnumber of values∈dataset

In this problem we calculated a sample mean. It helps to remember the calculation of a sample mean or a population mean is the same process.

Median:

First, must place the data values in ascending or descending order.

Second, then alternate from top to bottom by crossing out the data values.

Lastly, if you have an odd number of data values your remaining value will be the median. OR if you have an even number of data values in the data set you will have two data values remaining. Then add the two remaining values and divide by the number 2.

Mode:

The mode is the data value that occurs the most. And yes, you can have multiple modes occur in a data set.

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The data represents the cost of gas in 2005 for a sample of 20 vehicles:

13 15 15 8 16 20 28 19 18 15

21 23 30 17 10 16 15 16 20 15

Construct a frequency distribution for the data set using five classes:

Class Frequency Midpoint Relative Frequency

Boundaries Cumulative Frequency

Remember:

Class Width = Range

¿of classes = Max−Min¿of classes

Class Boundaries = (Lower limit of the2nd class )−(Upper limit of the 1 st class)

2

Frequency: Is how often a number occurs within a class.

Midpoint = (Upper limit of the1 st class )+(Lower limit of the 1 st class)

2

Relative Frequency = Frequency of classTotal Frequency

Cumulative Frequency: You add each frequency by the sum of the frequency of the preceding class.

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The data represents the hourly earnings (in dollars) of sample of 15 medical professionals.

39 36 30 27 26 24 28 35 39 60 50 41 35 32 51

Find the following:

a. Five-number summaryb. IQRc. Outliersd. Box-and-whisker plot

Remember:

a)

Min “Minimum” = lowest numbered sample in the set

Max “Maximum” = highest numbered sample in the set

Q3 “Quartile 3” = Is the middle value in the second half of the data set.

Q2 “Quartile 2” = median

Q1 “Quartile 1” = Is the middle value in the first half of the data set.

b)

IQR “Interquartile Range” = Q3 - Q1

c)

Outliers:

Must be calculated for “Outliers less than” and “Outliers more than.”

“Outliers less than” = Q1 – (1.5 * IQR)

“Outliers more than” = Q3 + (1.5 * IQR)

d)

Box-and-Whisker plot:

Plot all calculated data on a Box-and-Whisker plot

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11. The numbers of regular season wins for 10 football teams in a given season are given below. Determine the range, mean, variance, and standard deviation of the population data set.

2 10 15 4 15 8 15 9 4 7

The range is: _______

The population mean is: __________ (Round to the nearest tenth as needed.)

The population variance is: ________ (Round to the nearest hundredth as needed.)

The population standard deviation is: ___________ (Round to the nearest tenth as needed.)

Remember:

Range = Max – Min

µ “Population mean” = ∑ of all values∈data setnumber of values∈dataset

σ2 “Population Variance” = Take the square of the standard deviation.

σ “Population Standard Deviation” = The square root of the variance.

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The ages (in years) of a random sample of shoppers at a gaming store are shown. Determine the range, mean, variance, and standard deviation of the sample data set.

12 19 23 15 18 18 21 18 14 16

The range is: _______

The mean is: __________ (Round to the nearest tenth as needed.)

The variance is: ________ (Round to the nearest hundredth as needed.)

The standard deviation is: ___________ (Round to the nearest tenth as needed.)

Rememeber:

Range = Max – Min

x“Sample mean” = ∑ of all values∈data setnumber of values∈dataset

σ2 “Sample Variance” = Take the square of the sample standard deviation.

σ “Sample Standard Deviation” = The square root of the sample variance.

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The 7 Graphs commonly used in Statistics:

Qualitative Data Charts:

Are the Pie Chart and the Pareto Diagram.

Construct a Pie Chart from the data:

Type of degree in 2020

Number “in 1000’s”

Relative Frequency

Relative Frequency %

Angle in degrees

Associates 942Bachelors 1716Masters 731Doctoral 164

Remember:

Relative Frequency = Frequency

Total Frequency

Relative Frequency % = Frequency

Total Frequency * 100%

Angle in degrees = (Relative Frequency) * 360°

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Construct a Pareto Diagram from the data:

A company lost a total 34.4 billion.

Employee theft 15.1

Shoplifting 12.3

Vendor fraud 1.7

Administrative error 4.2

Unknown 1.1

Must remember: Qualitative data only applies to pie charts and Pareto

diagram.

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Also, remember: Qualitative data are the result of categorizing or describing attributes of a population. Qualitative data are also often called categorical data. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data.

Quantitative charts:

Are the Stem and Leaf Plot, Dot Plot, and Histogram:

Construct a Stem and Leaf Plot from the data:

23 55 23 54 24 52 26 33 31 32

Construct a Dot Plot from the data:

23 55 23 54 24 52 26 33 31 32

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Construct a Histogram from the data:

7 8 9 9 11 7 8 7 9 7 11 12 7

7 8 10 11 12 8 10 12 8 10 13

Remember: To mention skewness: To the left, to the right, uniform, and symmetric bell shaped.

Must remember: Quantitative data only applies to Stem and Leaf Plot, Dot

Plot, and Histogram.

Also, remember: Quantitative data are always numbers. Quantitative data

are the result

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of counting or measuring attributes of a population. Amount of money,

pulse rate, weight, number of people living in your town, and number of

students who take statistics are examples of quantitative data.

Paired Data Charts:

Are the Time Series Graph and the Scatter Plot:

Construct a Time Series Graph from the data:

Year Crime “murders”

1960 39

1965 52

1978 79

1983 75

1990 85

Remember: In other words, a Times Series Graph is a Scatter Plot with lines, yet time has to be

involved.

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Construct a Scatter Plot from the data:

Temperature in °F Sale of ice cream32 22543 25053 27572 34080 51095 730

Remember: To mention correlation coefficients: Positive correlation, negative correlation, and no correlation.

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Must remember: Paired data only applies to Time Series Graph and the Scatter plot.

Also, remember: Paired data in statistics, often referred to as ordered pairs, refers to two variables in the individuals of a population that are linked together in order to determine the correlation between them. In order for a data set to be considered paired data, both of these data values must be attached or linked to one another and not considered separately.

This method of paired data is used when a study hopes to compare two variables in individuals of the population to draw some sort of conclusion about the observed correlation. When observing these data points, the order of the pairing is important because the first number is a measure of one thing while the second is a measure of something entirely different.

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Empirical Rule:

For the data sets with distribution approximately symmetric and bell shaped, the standard deviation has these characteristics:

So, in other words, this is that about 68 percent of the values lie between z-scores of –1 and 1, about 95% of the values lie between z-scores of –2 and 2, and about 99.7 percent of the values lie between z-scores of -3 and 3. These facts can be checked, by looking up the mean to z area in a z-table for each positive z-score and multiplying by 2.

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Normal Distribution – Properties of Normal Distribution:

1) The mean, median, and mode are equal.2) The curve is bell shaped and symmetric.3) The total area under the curve is 1.4) The curve never touches the x-axis.5) The x−σ and x+σ , which is called the inflection points.

Remember: By definition, the requirements, for standard normal distribution is a normal distribution and must have a “mean of 0” and a “standard deviation of 1.”

Find the indicated area under the curve. Round to four decimal places.

1. To the left ofz = .08

2. To the right of z = -.65

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3. Between z = 0 andz = 2.86

4. To the left of z = - 1.96 and to the right of z = 1.96 “Hint: Remember area is always positive.”

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Tips and Tricks – Do’s and Don’ts of Calculator:

The Do’s:

Highly recommended hardware calculators!

Texas Instruments TI - 84 Plus CE: Casio FX-115 ES Plus:

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The 1st best software calculator!!! Also, it’s free! Get your 90-day free trail on Texas Instruments website.

https://education.ti.com/en/products/computer-software/ti-smartview-ce-for-84

TI SmartView CE for the TI – 84 Plus Family:

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The 2nd best software calculator!!! Also, it’s free! Get your 90-day free trail on Texas Instruments website.

https://education. ti .com/en/software/details/en/ FFEA90EE7F9B4C24A6EC427622C77D09/sda- ti -smartview- ti - 84 -plus

TI SmartView for the TI – 84 Plus Family:

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The Don’ts:

Do not buy these calculators…Texas Instruments TI-83 series calculators

Texas Instruments TI-30 series calculators

Texas Instruments TI-nSpire calculators

Casio FX-9750 GII calculator

Casio G series calculators

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