STATISTICAL ANALYSIS OF THE RESULTS FOR THE MEASUREMENT OF THE DENSITY OF

SUGAR

•Scientific experiments do not yield the same result every time they are repeated. •The variation of results is analyzed according to mathematical procedures that form part of the area of knowledge called statistics.•Statistical procedures are very clear for experiments which yield large amounts of data. •Many experiments, especially those done in college classes, yield only half a dozen values or so. Statistics is a little bit fuzzy in these situations.•The methods to use in these cases, where the number of values is small, are all approximate. •Different scientists use slightly different methods. In these cases, the most important thing is to show clearly which method was used to arrive at the final results.

STATISTICAL ANALYSIS … DENSITY OF SUGAR

Our Method

We are going to use a method based on the the average deviation of the means or A.D.M., for short. Others prefer to use the standard deviation of the means rather than the A.D.M. The reason we chose the A.D.M is that it is easier to calculate and its significance is easier to understand.

•We obtain the average (mean) of our results.•We obtain the A.D.M of our results.•We construct the interval: mean±3(A.D.M) •We then claim that we are 98% confident that the quantity we are measuring falls within that interval.

The Big Picture

The Little Picture

•We calculate the mean of the experimental results.

•We calculate the deviation of each value from the mean: [value - mean].

•We take the absolute value of each deviation.

•We average all the absolute values.•We now have the a.d.•To get the A.D.M, we divide the a.d. by the square root of the number of values.

•We construct the interval: mean±3(A.D.M.)•We make our 98% claim as indicated above.

STATISTICAL ANALYSIS … DENSITY OF SUGAR

ExampleLet’s say the measurements are:4.0, .5.0, 8.0, 6.0, 4.0, 5.0The mean of these measurements is

The a.d. (average deviation) is

For sample sizes from 4 to 20, only one digit in the a.d. is really significant. Therefore, the a.d. = 1

STATISTICAL ANALYSIS … DENSITY OF SUGAR

A.D.M stands for average deviation of the means. It is calculated by dividing the a.d divided by the square root of the number of values.

We conclude that here is a 98% chance that the value of the quantity we are measuring falls within 5.3±3(.5). In other words, it falls within the interval ranging from 3.8 to 6.8 .

STATISTICAL ANALYSIS … DENSITY OF SUGAR

Review in table form of the a.d and A.D.M. calculations

x m (x-m) |x-m|

4.0 5.3 -1.3 1.3

5.0 5.3 -.3 .3

8.0 5.3 2.7 2.7

6.0 5.3 .7 .7

4.0 5.3 -1.3 1.3

5.0 5.3 -.3 .3

Average of the fourth column = a.d = 1.1 A.D.M = a.d. divided by √6 = 0.5

Exercise: Find the mean, the average deviation, the average deviation of the means, and the 98% interval for this given set of data:

2.1, 2.3, 2.2, 2.4, 2.2, 2.4, 2.1 2.2 2.2, 2.1 and 2.22

x m (x-m) |x-m|

a.d = Interval = ( )A.D.M =

STATISTICAL ANALYSIS … DENSITY OF SUGAR

Properties of Matter

Extensive properties: depend on the amount of substance being considered.

DENSITY OF SUGAR

Intensive properties: they do not depend on the amount of material present