Data

• Quantitative data are numerical observation– age of students in a class. Age is quantitative data

because it quantifies the age of a person

• Qualitative data are categorical observation.– Marital status; married, single, divorced, widowed.

• Discrete quantitative data includes whole number data such as 2 cars, 3 children per family.

• Continuous data Data are mixed numbers or there is a possibility of more than one number between two whole numbers.

Continued

• Variable: assumes different value.– Gender is a variable which includes male, female

• grouping data– 5< classes<15

• Use Sturges rule to determine # of classes.• Use a bar graph to describe qualitative data• Use a clustered bar graph when each qualitative

data value contains at least two frequency column• Use a pie-chart when data are given in percent• Use a histogram when data are quantitative

Frequency Distribution.

• Relative frequency: frequency of a given data value/ total frequency.– Relative frequency is a fraction of the total

frequency.– 25% of the total number of students passed

the test.

• Cumulative frequency is used to show the number of observations below or above a certain value.– 5 students received marks less than 50

Measure of Central Tendency

• Extreme values:very high or low compared to the rest of data– Draw a box-plot to determine the number of extreme

values/outliers

• When there are no extreme values, use mean for the central tendency. X- is used to denote a sample mean

• When there are extreme values, use median for the central tendency

• Mode is the number that shows very often in a data set.

Measure of variation in data

• Range is the simplest measure of variation in data and is = Highest value -lowest value– Not a good measure because you are using only

two data of the data set.

• Standard deviation measures the dispersion of data from the mean of the data set.– Better than range because it used all data in the

data set.– Measure accurate when there are no extreme

values in the data set.

Mean of data

• When data are given without frequency column,• add all values of x and then divide by the total

number of data• For repeat data, multiply each data value by its

frequency and then add them. Divide the sum by the sum of all frequencies.

• For grouped data, get the class mark of each class and then multiply each class mark by its frequency. Add them all and didide by the total frequencies.

Median of data

• Arrange the data in ascending order and find the middle number.This applies to odd number of data. Formula: (n+1)/2 indicates the position of the number.

• When the number of data is even, find two middle numbers and get the arithmetic mean

• (n/2+(n/2+1)/2 indicates the positions of numbers.

• For median of repeat and grouped data, follow the formulas given in the class.

Mode

• Mode is the number that shows up very often in the data set

• For mode of repeat and grouped data, follow the formulas given in the class.

• If there are more than two modes in a given set of data, the sense of central tendency is obscured.

Quartiles

• First quartile value cuts of a lower-tail area of 25% of the distribution. If the first quartile mark of the test #1 is 60, 25 of the students received less than 60.(Q1)

• The second quartile cuts of the lower 50 of the distribution. (Q2)

• The third quartile separates the lower 75% of the distribution from the upper 25% of the distribution.

Skewness

• The distribution is skewed to the right when the right tail is longer than the left one. mean is > median> mode.

• The distribution is skewed to the left when the left tail is longer than the right one. Mean is <median<mode.

• The distribution is symmetric when the left side is the mirror image of the right.

Empirical Rule

• For the distribution that is symmetric and unimodal, about 68%of the data fall within one standard deviation of the mean.

• 95% of the data fall within 2 standard deviation of the mean.

• About 99.9 5 of the data fall within 3 standard deviation of the mean.