Measures of Position Section 3-3

Measures of position tell where a specific data value falls within the data set or its relative position in comparison with other data values. The most common measures of position are percentiles, deciles, and quartiles.

Objective Identify the position of a data value in a data set using various measures of position, such as percentiles, deciles, and quartiles. Make comparisons of data sets relative position using measures such as standard score.

Standard Score or Z-Score A standard score or z score is used when direct comparison of raw scores is impossible. A standard score or z score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation.

Z-Score or Standard Score Obtained by subtracting the mean from the value and dividing the result by the standard deviation. The symbol for the standard score is z.

Percentiles Divide the data set into 100 equal groups. Percentile Formula: You must put the data into order from lowest to highest.

Ex. Find the percentile rank of a score of 12 in the following data set.

Find a value corresponding to a given percentile where n = total number of values and p = percentile If c is not a whole number, round it up to the nearest whole number. Start at the lowest value and count over to that value, which would correspond to the percentile.

Ex. Find the value in the data set corresponding to the 30th percentile. 18, 15, 12, 6, 8, 2, 3, 5, 20, 10

Quartiles Divide a distribution into four groups. Percentile Ranks

Deciles Divide a distribution into 10 groups. Denoted by D1, D2, …, D9

Percentile Graph on males by weight and age Use the percentile graph shown to answer the following questions: If a man is 35 years old, what is his weight in kg if he is in the 50th percentile? If a man is in the 95th percentile and weighs approximately 255 pounds, how old is he? A 20 year old man weighs how much for the following percentiles: 5th ________ 25th _______ 50th _______ 75th _______ 95th _______

Interquartile Range (IQR) IQR is the difference between Q1 and Q3 and is the range of the middle 50% of the data.

Outliers An extremely high or low data value when compared to the rest of the data values.

Procedure for Identifying Outliers Step 1: Find Q1 and Q3. Step 2: Find the IQR (Q3 - Q1 ). Step 3 Multiply this by 1.5. Step 4: Subtract this value (step3) from Q1 and add it to Q3. Step 5: Check the data set for values less than Q1 – (1.5)IQR or larger than Q3 + (1.5)IQR.

Ex. Check the following data set for outliers Ex. Check the following data set for outliers. Be sure and give the range using Q1 – (1.5)IQR or larger than Q3 + (1.5)IQR. 5, 6, 12, 13, 15, 18, 22, 50

Exploratory Data Analysis 3-5

Exploratory Data Analysis The purpose of exploratory data analysis is to examine data in order to find out what information can be discovered. For example: Are there any gaps in the data? Can any patterns be discerned?

Boxplots and Five Number Summaries Boxplots are graphical representations of a five-number summary of a data set. The five specific values that make up a five-number summary are: The lowest value of data set (minimum) Q1 (or 25th percentile) The median (or 50th percentile) Q3 (or 75th percentile) The highest value of data set (maximum)