Statistics

- Mathematical method of dealing with the collection, presentation, analysis, and interpretation of

numerical facts and figures in order to draw conclusions and aid in

making decisions on the basis of such analysis.

Statistics

• Collection – process of obtaining numerical measurements or data.• Presentation or tabulation – organizing of data into tables, graphs or charts.• Analysis – process of formulating numerical description from the given data• Interpretation – drawing of valid conclusions from the analyzed data

Statistics

Variable Data

Discrete

Ungrouped Data

Grouped Data

Continuous

Ungrouped Data

Grouped Data

Raw Data – data presented in an array. It is a listing of the values as observed.Ordered Data – raw data is arranged in either in increasing or decreasing orderFrequency Data – frequency of occurrence of distinct data values or cells are givenUngrouped Data – enumeration of distinct dataGrouped Data – data grouped into cells or range of values

Frequency – is the number of times that a value is observed.

4.1 Frequency Tally SheetIt is a method that determines the

frequency value for the observed data or given cell. It provides a good idea of the general shape of the frequency data.

3 6 6 7 5 9 8 66 7 5 6 7 5 3 88 7 7 6 8 9 4 54 7 8 3 10 10 10 65 4 9 8 5 4 7 7

HouseholdSizes

FrequencyTally

FrequencyNumber

(f)

RelativeFrequency

Total(n):

Preparing Frequency Tally Sheets for Ungrouped Data

Relative Frequency

Household sizes of various residences in a barangay in Metro ManilaTable 4.1 – Frequency Tally Sheet

No. of Data (n)z No. of Classes (k)Under 50 5-7

50-100 6-10100-250 7-12Over 250 10-20

Grouped Data ParametersCells (k) – are also known as class intervals. It is the grouping of successive data values using a specific upper and lower limit.

Cell Width (c) – it is the numerical value distance between the limits of any two cell. If possible all intervals should have the same width for easier computation and interpretation.

(round down)

(round up)

Cell Boundaries – the cell or class boundaries are values one-half measurement unit more accurate that the observed data. Therefore, if the observed accuracy is one decimal, the boundaries are written in two decimal values. This is necessary so that no value can be observed exactly on the boundary.

Cell Midpoint – this is one half of the distance between the cell boundaries. Each midpoint is assumed to be the point of concentration of all values in the cell.

6.72 6.62 6.70 6.70 6.62 6.676.66 6.74 6.73 6.78 6.76 6.766.72 6.82 6.76 6.70 6.76 6.666.77 6.76 6.79 6.72 6.76 6.686.66 6.72 6.78 6.78 6.75 6.706.64 6.81 6.80 6.66 6.76 6.78

Diameters of the rivet heads in 1/100 of an inch

ClassBoundaries(LCB-UCB)

FrequencyTally

FrequencyNumber

(f)

RelativeFrequency

Midpoint()

Total (n):

MEASURES OF LOCATION(UNGROUPED DATA)A. Mean

The sample mean, denoted by, is the sum of the sample values divided by the sample size.

Where ;

= arithmetic mean = actual values

n = number of observationsAlso,

Where;

= frequency of each value

• Example1. An article in the Journal of Structural

Engineering (Vol. 115, 1989) describes an experiment to test the yield strength of circular tubes with caps welded to the ends. The first yields (in kN) are 96, 96, 102, 102, 102, 104, 104, 108, 126, 126, 128, 128, 140, 156, 160, 160, 164, and 170. Calculate the sample mean.

B. MedianMedian, represented by , is actually the middle value,

which divides the data set into two equal halves. Unlike the mean.

• The value of the median is found by ordering the data in increasing size and counting until the middle observation is reached.

• If the total number of observations is odd, is the center value.

• If it is an even value, is the average of the two middle values.

C. Mode• Mode, , is another measure of central

tendency which occurs at the most observed value of the variable.

• This is the value of the cell with the highest frequency.

• There should be at most 2 modes.

MEASURES OF VARIATION• Used to describe the variability or scattering of

data. This is used to gauge the reliability of averages as well as serve as a basis for control of variability itself.

A. Range• The range is computed as the difference

between the smallest and the largest value of the observation.

B. Standard DeviationThis is the most commonly used measure of

dispersion. It gives a numerical value in the units of the variable itself, to the clustering tendency of data.

Where xi = actual value of the observation

n = total number of observations = measure of average

fi = frequency of the observation