Variance and standard deviation

measures of central tendency and dispersion

Week 4

measures of central tendency, position and dispersion

Students will be able to:

i) Calculate of mean using formula and technology.Students should use mid-interval values to estimatethe mean of grouped data.ii) Calculate of standard deviation and variance ofthe sample using only technology, however handcalculations may enhance understandingiii) Determine the effect of constant changes on the original data.

A high school teacher (PhD Josué) has collected the test scores of 215 students on a Diagnostic Test of Maths:

01. initial problem

Calculate:a) Mean, median, and mode of the test scores.b) 1st, 2nd, 3rd quartiles and IQR.c) Variance and standard deviation of the test scores (consider the population measures).Graph:Graph a histogram and a Whisker-box plot (with and without outliers).Analyze:A) Based on the mean, median, and mode, describe the distribution of the test scores. Is it skewed left, skewed right, or symmetric?B) Identify any outliers in the data using the IQR method. How do these outliers affect the mean and median?C) What do the quartiles and interquartile range reveal about the distribution of the scores?D) How would you interpret the percentile rank of a student who scored 20 or more points?E) Based on your analysis, what conclusions can you draw about the students' overall performance on the exam?F) Using the standard deviation, discuss the spread of the scores. Are the scores clustered closely around the mean, or are they widely dispersed?

initial problem

02. review

Measures of Central Tendency

Measures of Position

Cumulative frequency graph

Parallel whisker-box plot

Bar graph with whisker-box plot

Whisker-box plot (w/outliers)

Whisker-box plot

Histogram

Bar Graph

Statistical Graphics

3. VARIANCE AND STANDARD DEVIATION

(we need to consider alternative measures of spread which take into account all data values of a data set!)

Remarks: 1) The standard deviation is the square root of the variance.2) The standard deviation is a non-resistant measure of spread.3) The IQR and percentiles are more appropriate tools for measuring spread if the distribution is considerably skewed.

3. VARIANCE AND STANDARD DEVIATION

examples

examples

We wants to compare the running speeds of boys and girls at his school. We randomly selects 10 boys and 10 girls, and records the time, in seconds, each person takes to run 1 lap on the red hallway.Which group:i generally runs faster ii has the greater spread of running speeds?¢ How could we improve the reliability of our findings?

examples

04. investigation

The histograms alongside show the times for the Girls 50 metre freestyle recorded by members of a requencyswimming squad.a) Copy and complete:b) Discuss the distributions of times for the boys and girls. What conclusion can you make?

exercises

1) The data set 4,6,9,a,3,b has a mean and mode of 6. Find the values of a and b given that a>b.2) Consider the data set: k—2, k, k+3, k+3.a) Show that the mean of the data set is equal to k+1.b) Suppose each number in the data set is increased by 2. Find the new mean of the data set in terms of k.

exercises

exercises