Data Presentation And Analysis 34k2y

Presentation and Analysis of Statistical Data

Statistics • A science of conducting studies to collect, organize, summarize, analyze, and draw conclusion from data; interpreting and presenting numerical data. • Can refer to the mere tabulation of numeric information as in reports of stock, market, transactions, or to the body of techniques used in processing or analyzing data.

Two Branches of Statistics Statistics can branches:

be

organized

into

two

• Descriptive statistics Concerned with collecting, organizing, presenting, and analyzing numerical data. • Inferential statistics Its main concern is to analyze the organized data leading to prediction or inferences.

Some Important Concepts • Data Data are the raw material which the statistician works. Data can be found through surveys, experiments, numerical records, and other modes of research.

• Population Refers to the groups or aggregates of people, objects, materials, events or thing of any form.

• Sample Consist of few or more of the population.

• Outliers An outlier is a data point that lies outside the overall pattern of a distribution.

Variables - types Statistical data or information can be gathered through different ways such as interviewing people, observing or inspecting items, using questionnaires and checklists. The characteristic that is being studied is called a variable. It varies from one person or thing to another. • Numerical values and are examples of quantitative variables. • Non-numerical values or attributes are examples of qualitative variables.

Dependent and Independent Variables A variable can be dependent or independent depending on its use. To predict the value of variable on the other, Independent variable is the predictor while the dependent variable is the variable whose value is being predicted.

Scales of Measurement of Data • Nominal Data Use numbers for the purpose of identifying name or hip in a group or category.

• Ordinal Data Connote ranking or inequalities in this type of data, numbers represents “greater than” or “less than” measurement, such as preferences or rankings.

• Interval Data Indicate an actual amount and there is equal unit of measurement separating each score, specifically equal intervals. The true zero is present.

• Ratio Data Similar to interval data but has an absolute zero and multiples are meaningful. It include all the usual measurement of length, height, weight, area, volume, density, velocity, money and duration.

Types of Data Primary Data - data collected directly by the researcher himself. These are first-hand or original sources. They can be collected through: 1. Direct observation or measurement (primary source of info). 2. By interview (questionnaires or rating scales). 3. By mail of recording or of recording forms. 4. Experimentation.

• Secondary Data - Information taken from published or unpublished materials previously gathered by other researchers or agencies such as book, newspapers, magazines; journals, published and unpublished thesis and dissertations.

Methods in Presenting Data • Textual Form - data in paragraph form. • Tabular Form systematic arrangement of data in rows and columns. • Graphical Form - a graph or chart is a device for showing numerical values in pictorial form. • Semi Tabular/Semi Tabular Form the combination of Textual and

Tabular Presentation • Used for summarization and tabulation of data. • Helps in analysis of relationships, trends, relative size of data. • Parts of a Table  Title and number  Caption ( column heading)  Stub (row heading)  Sources and footnotes (optional)

Graphical Presentation Types of graphs or Charts: • • • • • • • •

Line Plots Line Chart Bar Chart Stem and leaf plots Pie Charts Box Plots Pictograms Scatter Diagrams

Line Plots Consists of • a horizontal number line of the possible data values; • one X for each element in the data set placed over the corresponding value on the number line.

Works well when • • • •

the data is quantitative (numerical); there is one group of data (uni-variate); the data set has fewer than 50 values; the range of possible values is not too great.

Line Plot Example Suppose thirty people live in an apartment building. The ages of the residents are below. 58, 40, 54, 34, 47,

30, 47, 50, 40, 46

37, 47, 35, 46,

36, 39, 40, 49,

34, 54, 38, 47,

49, 47, 47, 35,

35, 48, 48, 48,

The graph is easier to create when the ages are placed in order from largest to smallest as the values will appear on the number line. 30, 37, 46, 48, 54,

34, 38, 47, 48, 58

34, 39, 47, 48,

35, 40, 47, 49,

35, 40, 47, 49,

35, 40, 47, 50,

36, 46, 47, 54,

Advantages of Line Plots • The plot shows all the data. • Line plots allow several features of the data to become more obvious, including any outliers, data clusters, or gaps.

• The mode is easily visible. • The range can be calculated quite easily from this data display.

Disadvantages of Line Plots • A line plot may only be used for quantitative (numerical) data. • A line plot is not efficient when the data is large and/or the the range is large.

Line Graph Consists of • paired values graphed as points on a plane defined by an x- and y-axis; • line segments connecting the graphed points (much like a dot-to-dot).

Works well when • the data is paired (bi-variate); • the data is continuous.

Line Graph Example

John weighed 68 kg in 1991, 70 kg in 1992, 74 kg in 1993, 74 kg in 1994, and 73 kg in 1995.

Advantages of Line Graphs • A line graph is a way to summarize how two pieces of information are related and how they vary depending on one another.

Disadvantages of Line Graphs • Changing the scale of either axes can dramatically change the visual impression of the graph

Bar Graph Consists of • bars of the same width drawn either horizontally or vertically; • bars whose length (or height) represents the frequencies of each value in a data set.

Works well when • the data is numerical or categorical; • the data is discrete; • the data is collected using a frequency table.

Bar Graph Example

Advantages of Bar Graphs • The mode is easily visible. • A bar graph can be used with numerical or categorical data.

Disadvantages of Bar Graphs • A bar graph shows only the frequencies of the elements of a data set.

Stem and Leaf Plot Consists of • Numbers on the left, called the stem, which are the first half of the place value of the numbers (such as tens values); • Numbers on the right, called the leaf, which are the second half of the place value of the numbers (such as ones values) so that each leaf represents one of the data elements.

Works well when • the data contains more than 25 elements; • the data is collected in a frequency table; • the data values span many “tens” of values.

Stem and Leaf Plot Example

The number of points scored by the Vikings basketball team this season: 78, 96, 88, 74, 63, 86, 92, 66, 72, 88, 83, 90, 67, 81, 85, 94. Writing the data in numerical order may help to organize the data, but is NOT a required step. 

63, 66, 67, 72, 74, 78, 81, 83, 85, 86, 88, 88, 90, 92, 94, 96

Separate each number into a stem and a leaf.  Since these are two digit numbers, the tens digit is the stem and the units digit is the leaf.  

The number 63 would be represented as Stem Leaf

Group the numbers with the same stems.  List the stems in numerical order.  Title the graph.

6

3

Points scored by the Vikings Stem

Leaf

6

3 6 7

7

2 4 8

8

1 3 5 6 8 8

9

0 2 4 6

Advantages of Stem and Leaf Plots • It can be used to quickly organize a large list of data values. • It is convenient to use in determining median or mode of a data set quickly. • Outliers, data clusters, or gaps are easily visible.

Disadvantages of Stem and Leaf Plots • A stem and leaf plot is not very informative for a small set of data.

Pie Chart Consists of • a circle divided into sectors (or wedges) that show the percent of the data elements that are categorized similarly.

Works well when • there is only one set of data (uni-variate); • comparing the composition of each part to the whole set of data.

Pie Chart Example Color White Black Gray Red B lue Other Total

Number 19 25 11 18 7 10 90

A proportion can be used to calculate the angle measure for each sector. Using white as the example, 19 white cars compare to the total of 90 in the same way that 76 degrees compares to the total degrees (360) in a circle.

Advantages of Pie Charts can be used for either numerical or categorical data.

• • shows a part to whole relationship.

Disadvantages of Pie Charts • Without technology, a pie chart may be difficult to make. Each percent must be converted to an angle by calculating the fraction of 360 degrees. Then the correct angle must be drawn. • A circle graph does not provide information about measures of central tendency or spread.

Box and Whisker Plot Consists of •

•

•

the “five-point summary” (the least value, the greatest value, the median, the first quartile, and the third quartile); a box drawn to show the interval from the first (25th percentile) to the third quartile (75th percentile) with a line drawn through the box at the median; line segments, called the whiskers, connecting the box to the least and greatest values in the data distribution.

Works well when • •

there is only one set of data (uni-variate); there are many data values.

Box and Whisker Plot Example Math test scores 80, 75, 90, 95, 65, 65, 80, 85, 70, 100.

Write the data in numerical order and find the five point summary.. median = 80 first quartile = 70 third quartile = 90 smallest value = 65 largest value = 100

Median

65, 65, 70, 75, 80, 80, 85, 90, 95, 100 Median of Lower Part, First Quartile

Place a point beneath each of these values on a number line.

Draw the box and whiskers and median line.

Median of Upper Part, Third Quartile

65 70

75 80 85 90

95 100

65 70

75 80 85 90

95 100

Box and Whisker Plot Example

The following set of numbers are the amount (arranged from least to greatest) of video games owned by each boy in the club. 18 27 34 52 54 59 61 68 78 82 85 87 91 93 100 68 is the median The median is the value exactly in the middle of an ordered set of numbers.

52 is the lower quartile

The lower quartile is the median of the lower half of the values (18 27 34 52 54 59 61). 87 is the upper quartile The upper quartile is the median of the upper half of the values (78, 82, 85, 87, 91, 93, 100).

Advantages of Box and Whisker Plots • Immediate visuals of a box-and-whisker plot are the center, the spread, and the overall range of distribution. • Box plots are useful for comparing data sets, especially when the data sets are large or when they have different numbers of data elements.

Disadvantages of Box and Whisker Plots • It shows only certain statistics rather than all the data. • Since the data elements are not displayed, it is impossible to determine if there are gaps or clusters in the data.

Scatter Plot Consists of • paired data (bi-variate) displayed on a twodimensional grid.

Works well when • multiple measurements are made for each element of a sample.

Additional Notes about Scatter plots • If the relationship is thought to be a causal one, then the independent variable is represented along the x-axis and the dependent variable on the y-axis • A scatter plot can show that there is a positive, negative, constant, or no relationship (correlation) between the variables. – Positive: As the value of one variable increases, so does the other. – Negative: As the value of one variable increases, the other decreases. – Constant: As the value of one variable increases (or decreases), the other remains constant. – No relationship: There is no pattern to the points.

Scatter plot Example

Advantages of Scatter plots • A scatter plot is one of the best ways to determine if two characteristics are related. • A scatter plot may be used when there are multiple trials for the same input variable in an experiment.

Disadvantages of Scatter plots • When a scatter plot shows an association between two variables, there is not necessarily a cause and effect relationship. Both variables could be related to some third variable that explains their variation or there could be some other cause. Alternatively, an apparent association could simply be a result of chance

Histogram Consists of • equal intervals marked on the horizontal axis; • bars of equal width drawn for each interval, with the height of each bar representing either the number of elements or the percent of elements in that interval. (There is no space between the bars.) Works well when • data elements could assume any value in a range; • there is one set of data (uni-variate); • the data is collected using a frequency table.

Histogram Example

Advantages of Histograms • A histogram provides a way to display the frequency of occurrences of data along an interval.

Disadvantages of Histograms • The use of intervals prevents the calculation of an exact measure of central tendency.

Frequency Polygon Consists of It is a line chart that is constructed by plotting the frequencies and class mark and connecting the plotted pointed by means of a straight line; the polygon us closed by considering an additional class at each end and each end of the lines are brought down to the horizontal axis at the mid point of the additional classes.

Frequency Polygon (Data) C.I.

f



X



f/n

sector

80-89

1

40

84.5

1

0.0025

9percen t

70-79

4

39

74.5

5

0.1000

36

60-69

13

35

64.5

18

0.3250

117

50-59

13

22

54.5

31

0.3250

117

40-49

4

9

44.5

35

0.1000

36

30-39

3

5

34.5

38

0.0750

27

20-29

1

2

24.5

39

0.0250

9

10-19

1

1

14.5

40

0.0250

9

n=40

∑rf=1

Advantages of Frequency Polygon • A Frequency Polygon provides a way to display the frequency of occurrences of data along an interval. • Denotes Class Mark

Disadvantages of Frequency Polygon • The use of intervals prevents the calculation of an exact measure of central tendency.

Ogive Consists of • Graph of a cumulative frequency distribution and sometimes called a cumulative frequency distribution graph Works well when • Less Than And More Than intervals are required.

Ogive

Advantages of Ogive • A Ogive provides a way to display the cumulative frequency of occurrences of data along an interval.

Disadvantages of Ogive • The use of Cumulative intervals prevents the calculation of an exact frequency and measure of central tendency.

MEASURES OF CENTRAL TENDENCY -single number represent the given data. 1. Mean – average value of the given data. - not appropriate measures of central tendency if there is outlier.

2. Median – divide the distribution into two equal parts (upper 50% and the lower 50%) 3. Mode – the most frequent occuring data. - nominal value/part.

Mean

UNGROUPED DATA 25 32 41 58 78 9 5 105 110 112 112 115

Mean = ∑ X n

= 883 11 = 80.2727

GROUPED DATA Mean = ∑fx n = 2250 40 Mean = 56.25

Short Cut Method Mean = AM (∑fd/n) i = 64.5+ (-33/40)10 = 64.5-8.25 Mean = 56.25

Median

Median = LL+(n/2
Mode

Mode= LL + (∆1/ ∆1+ ∆2) h *where ∆1 = difference between the modal class and the next lower score. ∆2 = difference between the modal class and the next upper score.

= 50 + (13/13 +13)10 Mode = 55

QUARTILE (Q)

Three Quartiles • Q1- 25%

1/4

First Quartile

• Q2- 50%

1/2

Second Quartile/ Median

• Q3- 75%

3/4

Third Quartile

Q1 = LL + n/4 -
Q3 = LL + 3n/4 -
PERCENTILE (P) • • • • •

P1- 1/100 P2- 2/100 P3- 3/100 P4- 4/100 P5- 5/100 ….. • P99- 99/100

P23 = LL + 0.23n -
Measures of Variability •Measure of the scatteredness of a particular data in a given data set. •Average of distance •Used to determine the reliability of average values and control variation

Following are the measures of variation : • Range - Difference between values of the two extreme items of a series • Interquartile Range – Difference between third and First Quartile • Mean Average Deviation – the arithmetic mean of the absolute differences of the values from their average. • Standard Deviation – positive Square root of the average of Squares of Deviations of values from their means.]

Range

R = XL – Xs Example :Age of a sample of 10 subjects from a population are as follows: 42, 28, 28, 61, 31, 23, 50, 34, 32, 37. Range = 61 – 23 = 38. Coeff of Range = (XL – Xs)/ (XL + Xs) = (61 – 23)/ (61 +23) = 0.452

InterQuartile Range(IQR)

IQR = (Q3 – Q1) Semi Quartile Range = (Q3 – Q1)/2 Coeff of Quartile Deviation = (Q3 – Q1)/ (Q3 + Q1)

Mean Deviation

• For Ungrouped Data : MAD = ∑|x-x| n • For Grouped Data : MAD = ∑fi|xi-x| ∑fi Coeff of MAD : = (MAD/ Mean)

Standard Deviation

For Ungrouped Data: σ = √∑(Xi – X)2/n

For Grouped Data: σ = √∑ fi(Xi – X)2/∑fi Coeff of Variation = (σ/ X) X 100 Variance = σ2