Data Collection Mean, Mode, Median, & Range Standard Deviation Mathematical Terms Related to a Group of Numbers Data Collection Mean, Mode, Median, & Range Standard Deviation

(answer is a counted numbers for sig figs are not an issue) Mean = the average Consider your measurements as a set of numbers. Add together all the numbers in your set of measurements. Divide by the total number of values in the set Ages of cars in a parking lot: 20 years +10 years +10 years +1 year+15 years +10 years= 66 years/6 cars = 11years (answer is a counted numbers for sig figs are not an issue) Note: the mean can be misleading (one very high value and one very low)

Median The number in the middle Put numbers in order from lowest to highest and find the number that is exactly in the middle 20, 15, 10, 10, 10, 1 Since there is an even number of values the median is 10 years (average of the 2 middle values)

Or to determine the Median: 20 years, 15 years, 13 years, 11 years, 7 years, 5 years, 3 years With an odd number of values the median is the number in the middle or 11 years

Mode Number in data set that occurs most often 20, 15, 10, 10, 10, 1 Sometimes there will not be a mode 20, 17, 15, 8, 3 Record answer as “none” or “no mode” – NOT “0” Sometimes there will be more than one mode 20, 15, 15, 10, 10, 10, 1

Range The difference between the lowest and highest numbers 20, 15, 10, 10, 1 20-1 = 19 years The range tells you how spread out the data points are.

Example: The mean of four numbers is 50.5 101 99 1 1 What is the median? What is the mode?

Measured Values When making a set of repetitive measurements, the standard deviation (S.D.) can be determined to indicate how much the samples differ from the mean Indicates also how spread out the values of the samples are

Standard Deviation The smaller the standard deviation, the higher the quality of the measuring instrument and your technique Also indicates that the data points are also fairly close together with a small value for the range. Indicates that you did a good job of precision w/your measurements.

A high or large standard deviation Indicates that the values or measurements are not similar There is a high value for the range Indicates a low level of precision (you didn’t make measurements that were close to the same) The standard deviation will be “0” if all the values or measurements are the same.

Formula for Standard Deviation range (highest value – lowest value) N = number of measured values As N gets larger or the more samples (measurements, scores, etc.), the reliability of this approximation increases = =

The Standard Deviation would be: 22.5 mL, 18.3 mL, 20.0 mL, 10.6 mL The Standard Deviation would be: range (highest value – lowest value) Range = 22.5 – 10.6g = 11.0 mL N = 4 11.0g = = 5.95 mL = = 6.0 mL S.D. = 17.9 ± 6.0 mL (expressed to the same level of precision as the mean)

Example The results of several masses of an object weighed by 5 different students on a multi-beam balance 11.36g, 11.37g, 11.40g, 11.38g & 11.39g Average = 56.90g ÷ 5 = 11.38g SO:

The Standard Deviation would be: 11.36g, 11.37 g, 11.40 g, 11.38 g & 11.39g The Standard Deviation would be: range (highest value – lowest value) Range = 11.40g – 11.36g = 0.04g N = 5 0.04g = = 0.0178 g = = 0.02 g

The Standard Deviation from the mean is: Mean = 11.38 g SD = 0.02 g Expressed as 11.38g ± 0.02 This tells us that the measurements were within two hundredths of the mean – either less than the mean or greater than the mean.

~Review~ Mean – average Mode – which one occurs most often Median – number in the middle Range – difference between the highest and lowest values Standard Deviation – The standard deviation of a set of data measures how “spread out” the data set is. In other words, it tells you whether all the data items bunch around close to the mean or is they are “all over the place.”

Graphical Analysis

The graphs show three normal distributions with the same mean, but the taller graph is less “spread out.” Therefore, the data represented by the taller graph has a smaller standard deviation.