Statistics with TI-Nspire™ Technology

Module E

Lesson 1: Elementary concepts

In this lesson you will:

• Study numerical and categorical data.• See the difference between a sample and a population.• Explore statistics to characterize the central tendency and the

dispersion of a dataset.• Learn to make illustrations of a dataset.

2 | Lesson E.1

TI-Nspire™ Technology

Statistics

• Statistics is a branch of mathematics that deals with the collection and summarization of data. • Data are observations (such as measurements or survey responses) that have been collected. • Statisticians often collect data from small portions of a large group or population in order to determine information about the group.

sample population

3 | Lesson E.1

Numerical and categorical data

• Numerical Data are collected on numerical variables. Examples are average temperature of the day, number of meals served per day, weight or length of a baby.

• Categorical Data represent the characteristics of objects or individuals in groups (categories). Examples are race, gender, hair-color, and educational level.

4 | Lesson E.1

One-variable statistics

• Descriptive statistics are used to describe the basic features of the data. They provide simple summaries about the sample and the measures.

• One-variable statistics involves the examination across cases of one variable at a time. • There are three major characteristics of a single variable that we tend to look at:

• the distribution• the central tendency• the dispersion

5 | Lesson E.1

Distribution of the data

• The distribution is a summary of the frequency of individual values or ranges of values for a variable.

• Pie Charts• Dot Plots• Line Plots• Scatter Plots• Box Plots• Histograms• Bar Graphs

• Visual illustrations are an important part of statistics.• Depending on the type of data, you can use

6 | Lesson E.1

Central tendency

26 31 34 36 36 36 36 37 37 37

37 37 37 37 37 37 37 38 38 38

38 38 38 38 38 38 38 38 38 38

38 38 38 38 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 41 41 41 41 41 41

7 | Lesson E.1

Central tendency: the median

26 31 34 36 36 36 36 37 37 37

37 37 37 37 37 37 37 38 38 38

38 38 38 38 38 38 38 38 38 38

38 38 38 38 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 41 41 41 41 41 41

8 | Lesson E.1

The first quartile

26 31 34 36 36 36 36 37 37 37

37 37 37 37 37 37 37 38 38 38

38 38 38 38 38 38 38 38 38 38

38 38 38 38 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 41 41 41 41 41 41

9 | Lesson E.1

The third quartile

26 31 34 36 36 36 36 37 37 37

37 37 37 37 37 37 37 38 38 38

38 38 38 38 38 38 38 38 38 38

38 38 38 38 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 41 41 41 41 41 41

10 | Lesson E.1

The outliers

26 31 34 36 36 36 36 37 37 37

37 37 37 37 37 37 37 38 38 38

38 38 38 38 38 38 38 38 38 38

38 38 38 38 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 39 39 39 39 39 39 39 39 39

39 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 40 40 40 40 40 40

40 40 40 40 41 41 41 41 41 41

11 | Lesson E.1

The box-plot

26 31 34

36

38 40

41

39

12 | Lesson E.1

Mean

• The mean is probably the most commonly used method of describing central tendency.

• The sample-mean is the average of all the values in the sample. • To compute the mean you add up all the values in the sample and

divide by the number of values.

1

1 n

ii

x xn =

= ∑

13 | Lesson E.1

Dispersion

• Dispersion refers to the spread of the values around the central tendency. There are two common measures of dispersion, the range and the standard deviation.

• The range is simply the highest value minus the lowest value. In our example distribution, the highest value is 41 and the lowest is 26, so the range is 41 - 26 = 15.

• The Sample Standard Deviation is a more accurate and detailed estimate of dispersion because an outlier can greatly exaggerate the range. The standard deviation shows the relation that the values in the dataset have to the mean of the dataset.

14 | Lesson E.1

( )2

1

11

n

ii

s x xn =

= −− ∑

The normal probability density function (PDF)

• You can also define a mean and a standard deviation of the population: µ (mean) and (standard deviation).

• Some populations have a normal distribution. They can be represented by a Gaussian curve.

μ = 3,3

3,3

σ = 0.58

15 | Lesson E.1

Relation between two lists

• Scatter plot or Line plot with two variables on two axes.• Regression curve.

16 | Lesson E.1

Categorical data

• Categorical variables represent types of data which may be divided into groups.

• Three different types of graphs: dot chart, pie chart or bar chart.

17 | Lesson E.1

In this lesson you learned:

The difference between numerical and categorical data. That you need different graph-types, depending on the type of data. How to calculate statistics to characterize the central tendency and the

dispersion of a dataset. That the distribution of a sample looks like the distribution of the

population. There is a difference between the mean and standard deviation of a

sample and the mean and standard deviation of the population. That some populations can be described by a Gaussian curve. That you can calculate percentages by integrating a density function. How to graph a relationship between two variables and add a regression

curve.

18 | Lesson E.1

Congratulations!

You have just finished lesson E.1!

19 | Lesson E.1