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Histograms and Distributions
HISTOGRAMS AND DISTRIBUTIONS
Histograms and Distributions
Suppose you want to know if athletes have faster reflexes than
non-athletes?
In order to get as close to the answer to this question as
possible you decide to run an experiment:
Using a web-based program
you measure the reaction
times of 25 athletes and 25
non-athletes under controlled
conditions.
Histograms and Distributions
Frequency refers to how often a particular value appears in the data:
Reaction Time
frequency
230
1
231
0
232
0
233
2
234
0
235
0
236
0
237
0
Etc…
Histograms and Distributions
A Histogram is a plot of frequency:
Histogram
2.5
1.5
Series1
1
0.5
0
200
207
214
221
228
235
242
249
256
263
270
277
284
291
298
305
312
319
326
333
340
frequency
2
Time (ms)
This is a weak attempt at making an
informative histogram…why?
Histograms and Distributions
bins
It would be more informative to place the
data into intervals called bins.
You choose the appropriate bin size. The
above bins have an interval of 10.
Histograms and Distributions
If the bin intervals are too small, the
histogram will be too spread out…
Histogram
2.5
1.5
Series1
1
0.5
0
200
206
212
218
224
230
236
242
248
254
260
266
272
278
284
290
296
302
308
314
320
326
332
338
frequency
2
Time (ms)
The bins above have an interval of 1…
Histograms and Distributions
If the bin intervals are too large, the
information will be too clumped:
Histogram
13.2
13
frequency
12.8
12.6
12.4
Series1
12.2
12
11.8
11.6
11.4
201-280
281-360
Time (ms)
The bins above have an interval of 80…
bins
Histograms and Distributions
Let’s go back to a bin interval of 10 and
look at the resulting histogram…
Histograms and Distributions
Histogram
4.5
4
frequency
3.5
3
2.5
2
Series1
1.5
1
0.5
0
Time (ms)
This is a decent choice. Remember that
all intervals must have the same size…
Histograms and Distributions
Histogram
4.5
4
frequency
3.5
3
2.5
2
Series1
1.5
1
0.5
0
Time (ms)
SAMPLE SIZE: Currently the sample size is only
25 students in the non-athlete group. Let’s see
what happens to our histogram as more data is
collected (sample size increases)…
Histograms and Distributions
SAMPLE SIZE: The sample size is now 73 students. Let’s compare the before
and after histograms…
Histogram
non-athletes
0
1
2
2
1
2
2
6
12
17
15
9
4
0
73
18
16
frequency
bin
200-210
210-220
221-230
231-240
241-250
251-260
261-270
271-280
281-290
291-300
301-310
311-320
321-329
330-339
sample size
14
12
10
8
Series1
6
4
2
0
Time (ms)
Histograms and Distributions
SAMPLE SIZE: The sample size is now 73 students. Let’s compare the before
and after histograms…
Histogram (after)
Histogram (before)
4.5
18
4
2.5
2
Series1
1.5
1
14
12
10
8
4
330-339
321-329
311-320
301-310
291-300
281-290
271-280
261-270
251-260
241-250
231-240
221-230
0
210-220
2
0
Time (ms)
Series1
6
0.5
200-210
frequency
3
frequency
16
3.5
Time (ms)
Histograms and Distributions
We can imagine that our intervals are infinitely small and our sample size is
infinitely large, which will result in the formation of a smooth curve:
Histogram
18
16
frequency
14
12
10
8
Series1
6
4
2
0
Time (ms)
Histograms and Distributions
This curve is known as a Normal Distribution or Bell-Shaped Curve… It
represents the probability of getting a data point in a given range or data.
Histogram
18
16
frequency
14
12
10
8
Series1
6
4
2
0
Time (ms)
Histograms and Distributions
For example, the probability of you next measurement being between 261 and
341 is near 100%. Likewise, the probability of your next measurement being
between 261 and 300 is around 50% as this is half the area under the curve.
Histogram
18
16
frequency
14
12
10
8
Series1
6
4
2
0
Time (ms)
Histograms and Distributions
What is the probability of your next data measurement being 291.34544 ms?
Near ZERO since this is only tiny fraction of the curve.
Histogram
18
16
frequency
14
12
10
8
Series1
6
4
2
0
Time (ms)
Descriptive Statistics
DESCRIPTIVE STATISTICS
Descriptive
Histograms
and Statistics
Distributions
Measures of Central Tendency
1. The MEAN:
This should be something you can already perform on a data set. Sum the
numbers and divide this by the number of numbers you have.
It can by expressed mathematically by the equation above where x is a random
variable that you are measuring and n is the number of measurements you have
made.
Descriptive
Histograms
and Statistics
Distributions
Measures of Central Tendency
2. The MEDIAN:
This is simply the value in a data set that separates the higher half of a sample
from the lower half.
For example, in the sample to the right, the value that
separates the higher and lower halves of data is
291ms, which is the median.
Reaction
Time (ms)
265
273
286
291
293
Just arrange the data from highest to lowest or vice
versa and find the central number…
300
330
Descriptive
Histograms
and Statistics
Distributions
Measures of Central Tendency
2. The MEDIAN:
This is simply the value in a data set that separates the higher half of a sample
from the lower half.
What if there is an even number of data points like
shown on the right?
Just average the two central measurement. In this
case you average 286 and 291 to get a median of 289.
Reaction
Time (ms)
265
273
286
292
293
300
Descriptive
Histograms
and Statistics
Distributions
Careful with the MEAN and MEDIAN
For example, a college boasts that the average starting salary of their last years
graduating class was $362,000 per year. This sounds quite impressive…
However, what they did not tell you was that the class size was 30 students of
which 28 started at $30,000 a year and one student was first round draft pick in
the NFL making approximately $10,000,000 per year.
Histogram
An outlier can be seen in the
histogram to the right of our athlete
data…perhaps the person blinked
while the reaction time was being
measured.
18
16
14
frequency
Such a data point ($10,000,000 per
year) can be considered an outlier,
which is a data point much higher
or lower than the rest of the data
points.
12
10
8
Series1
6
4
2
0
Time (ms)
Descriptive
Histograms
and Statistics
Distributions
Careful with the MEAN and MEDIAN
For example, a college boasts that the average starting salary of their last years
graduating class was $362,000 per year. This sounds quite impressive…
However, what they did not tell you was that the class size was 30 students of
which 28 started at $30,000 a year and one student was first round draft pick in
the NFL making approximately $10,000,000 per year.
What is the median of this data set? $30,000
The median is far less sensitive to outliers than the mean.
Descriptive
Histograms
and Statistics
Distributions
Careful with the MEAN and MEDIAN
So should we be focusing on the median more than the mean????
No. Generally speaking, the mean is TYPICALLY a far more accurate measurement in
terms of central tendency than the median when outliers have been dealt with.
To convince yourself, try this exercise from Seeing Statistics (www.seeingstatistics.com):
The median is more resistant to extreme, misleading data values so it would seem to be the clear choice.
However, we also need to consider accuracy. Is the median or the mean more likely to be close to the true value?
To evaluate the relative accuracy of the median and the mean, let's consider how they do when we know the true
center of the data. Suppose that the only possible scores are the whole numbers between 0 and 100.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
The center of these 101 numbers, whether we use the median or the mean, is 50. What if we were to select five
numbers randomly from this set of 101 and calculate the median and mean of those five numbers? Would the
median or the mean be closer to what we know is the true value of 50?
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
1. The RANGE:
This is simply the length of the smallest interval containing all of the data
For example, the range of the data to the right would
be…
265 ms to 300 ms
Reaction
Time (ms)
265
273
286
However, the range suffers from the same drawbacks
as the mean and even more so in terms of describing
data due to, once again, … outliers.
292
293
300
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
1. The RANGE:
This is simply the length of the smallest interval containing all of the data
Calculate the range now with the addition of one new
measurement that happens to be an outlier:
265 ms to 734 ms
Reaction
Time (ms)
265
273
286
The range is more sensitive to outliers than the mean
because with a large sample size, the effect on the
mean is diluted.
292
293
300
734
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
2. The INTERQUARTILE RANGE:
The interquartile (between quarters) range is one way around the outlier issue.
This value is calculated by first splitting the data up into four sections (quarters)
from low to high with the same number of data points in each section as shown
below:
The interquartile range is the range between the number that defines the upper end
of Quarter 1 (Q1) and the lower end of Quarter 3 (Q3)…let’s look at an example.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
2. The INTERQUARTILE RANGE:
Calculate the interquartile range of this data:
A. Find the median
268 ms, the 13th value
B. Now find the median of the first half of the data
excluding the 13th value
(231 + 231) / 2 = 231 ms = Q1
C. Find the median of the second half of the data
excluding the 13th value
(290 + 294) / 2 = 292 ms = Q3
D. The interquartile range is 231 ms to 292 ms.
It is also sometimes stated as Q3 – Q1, which
would be 61 ms in this case.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
2. The INTERQUARTILE RANGE:
If you start with an even number of data points as
shown to the right then…
Split the data in half and find the median of each
half. In this case one would split the data between
values 12 and 13.
A. The median of the top half is 231 ms again.
B. The median of the bottom half is (287 + 290)/2 =
288.5 (289) ms.
C. The interquartile range is 231 ms to 289 ms.
It is also sometimes stated as Q3 – Q1, which
would be 58 ms in this case.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
The Standard Deviation is simply a value
describing the distance from the mean in BOTH
directions that will encompass 68% of your data
on average.
Therefore, σ is a direct measure of the spread of
your data…let’s look at a quick example.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
This histogram shows blood
pressure data for a large sampling
of adult males.
The mean is around… 82 mmHg
σ is around…10 mmHg
What does this mean?
It means that between 82 +/- 10
mmHg (between 72 and 92 mmHg)
falls 68% of the data points.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Therefore, the more spread out
your data is…
…the greater the value of σ.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
To take it a step further, two standard
deviations away from the mean on
both sides (+/- 2σ) will encompass…
95% of the data.
Likewise, +/- 3σ will encompass
99.7% of the data.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
How does one calculate the Standard Deviation (σ)?
Let’s go back to our athlete/non-athlete reaction time
data to see how this is done starting with the nonathlete sample…
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Where should we begin?
−
By calculating the mean (X)…
278.5 ms
Now what? (think about what σ tells us)
It describes the spread of the data (or width of the normal
distribution / bell-shaped curve). Therefore, it is only logical to
find how far away all of your data is from the mean…
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Non-Athletes
Individual
Reaction Time (ms)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
X
210
225
233
233
247
256
257
268
270
274
276
278
282
286
287
295
298
298
300
305
307
311
314
324
329
−
X-X
-68.52
-53.52
-45.5
-45.5
-31.5
-22.5
-21.5
-10.5
-8.5
-4.5
-2.5
-0.5
3.5
7.5
8.5
16.5
19.5
19.5
21.5
26.5
28.5
32.5
35.5
45.5
50.5
210-278.5
225-278.5
233-278.5
233-278.5
…
− =278.5 ms
X
−
X – X = the mean minus the measured value
Now we are starting to get an idea about how
spread out the data is from the mean, which
is what σ is all about.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Non-Athletes
Individual
Reaction Time (ms)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
X
210
225
233
233
247
256
257
268
270
274
276
278
282
286
287
295
298
298
300
305
307
311
314
324
329
−
X-X
-68.52
-53.52
-45.5
-45.5
-31.5
-22.5
-21.5
-10.5
-8.5
-4.5
-2.5
-0.5
3.5
7.5
8.5
16.5
19.5
19.5
21.5
26.5
28.5
32.5
35.5
45.5
50.5
−
(X - X)2
4694.9904
2864.3904
2070.25
2070.25
992.25
506.25
462.25
110.25
72.25
20.25
6.25
0.25
12.25
56.25
72.25
272.25
380.25
380.25
462.25
702.25
812.25
1056.25
1260.25
2070.25
2550.25
The next step is to…
−
square all of the differences (X - X)2
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Non-Athletes
Individual
Reaction Time (ms)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
X
210
225
233
233
247
256
257
268
270
274
276
278
282
286
287
295
298
298
300
305
307
311
314
324
329
−
X-X
-68.52
-53.52
-45.5
-45.5
-31.5
-22.5
-21.5
-10.5
-8.5
-4.5
-2.5
-0.5
3.5
7.5
8.5
16.5
19.5
19.5
21.5
26.5
28.5
32.5
35.5
45.5
50.5
−
(X - X)2
4694.9904
2864.3904
2070.25
2070.25
992.25
506.25
462.25
110.25
72.25
20.25
6.25
0.25
12.25
56.25
72.25
272.25
380.25
380.25
462.25
702.25
812.25
1056.25
1260.25
2070.25
2550.25
Then…
You, for the most part, average the
squares:
−
(X - X)2 / n-1
The reason one uses n-1 is to account for
sample size. If n is large you are
essentially dividing by n and averaging. If
n is small like a sample size of n=3, then n1 makes a large difference in the resulting
prediction of σ.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Non-Athletes
Individual
Reaction Time (ms)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
X
210
225
233
233
247
256
257
268
270
274
276
278
282
286
287
295
298
298
300
305
307
311
314
324
329
−
X-X
-68.52
-53.52
-45.5
-45.5
-31.5
-22.5
-21.5
-10.5
-8.5
-4.5
-2.5
-0.5
3.5
7.5
8.5
16.5
19.5
19.5
21.5
26.5
28.5
32.5
35.5
45.5
50.5
−
(X - X)2
4694.9904
2864.3904
2070.25
2070.25
992.25
506.25
462.25
110.25
72.25
20.25
6.25
0.25
12.25
56.25
72.25
272.25
380.25
380.25
462.25
702.25
812.25
1056.25
1260.25
2070.25
2550.25
Then…
You essentially average the squares:
−
(X - X)2 / n-1 =
998.2
This number is known as the variance and
is directly related to the spread of your
data.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Non-Athletes
Individual
Reaction Time (ms)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
X
210
225
233
233
247
256
257
268
270
274
276
278
282
286
287
295
298
298
300
305
307
311
314
324
329
−
X-X
-68.52
-53.52
-45.5
-45.5
-31.5
-22.5
-21.5
-10.5
-8.5
-4.5
-2.5
-0.5
3.5
7.5
8.5
16.5
19.5
19.5
21.5
26.5
28.5
32.5
35.5
45.5
50.5
−
(X - X)2
4694.9904
2864.3904
2070.25
2070.25
992.25
506.25
462.25
110.25
72.25
20.25
6.25
0.25
12.25
56.25
72.25
272.25
380.25
380.25
462.25
702.25
812.25
1056.25
1260.25
2070.25
2550.25
One more step to get σ…
Square root the “average” to go back:
√
−
(X - X)2 / n-1 =
31.6
This is the standard deviation (σ). What
does this number mean?
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Non-Athletes
Individual
Reaction Time (ms)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
X
210
225
233
233
247
256
257
268
270
274
276
278
282
286
287
295
298
298
300
305
307
311
314
324
329
−
X-X
-68.52
-53.52
-45.5
-45.5
-31.5
-22.5
-21.5
-10.5
-8.5
-4.5
-2.5
-0.5
3.5
7.5
8.5
16.5
19.5
19.5
21.5
26.5
28.5
32.5
35.5
45.5
50.5
−
(X - X)2
4694.9904
2864.3904
2070.25
2070.25
992.25
506.25
462.25
110.25
72.25
20.25
6.25
0.25
12.25
56.25
72.25
272.25
380.25
380.25
462.25
702.25
812.25
1056.25
1260.25
2070.25
2550.25
It means that ACCORDING TO THE
CURRENT DATA, 68% of future data
collected should fall between 279 +/- 31.6.
Read the red text above over and over…
as your stats are only as good as your
data. Use common sense.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Standard deviation formula (what we just did):
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Your turn, athlete data…
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Your turn, athlete data…
264.4 +/- 30.6 ms
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Summary of current data:
athletes
Mean +/- σ
Nonathletes
264 +/- 30.6 279 +/- 31.6
What does it mean? … patients
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
The significance of the standard deviation:
The graph on the right
shows two data sets
having the SAME mean.
What is different then?
The blue data set has a greater
spread and therefore a larger σ.
Which data set would you
prefer (if you had a choice)?
The red one as there is less noise / variability. Variability is an inevitable limitation
in the methods we use to observe nature. It is your job to make as precise a
measurement as possible thereby limiting the variability.
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Compare the histograms of non-athletes to athletes:
Histogram
Histogram
4.5
4
4
3.5
3.5
3
2.5
Series1
2
1.5
frequency
3
2.5
Series1
2
1.5
1
1
0.5
0.5
Non-athletes
Better yet, overlay the histograms…
reaction time (ms)
Athletes
330-339
321-329
311-320
301-310
291-300
281-290
271-280
261-270
251-260
241-250
231-240
221-230
330-339
321-329
311-320
301-310
291-300
281-290
271-280
261-270
251-260
241-250
231-240
221-230
210-220
200-210
reaction time (ms)
210-220
0
0
200-210
frequency
4.5
Histograms and Distributions
Compare the histograms of non-athletes to athletes:
Mean +/- σ
athletes
Nonathletes
264 +/- 30.6
279 +/- 31.6
Number of students (frequency)
4.5
4
3.5
3
2.5
Non-athletes
Series1
2
Series2
Athletes
1.5
1
Q: Is there really a difference
between these two groups???
0.5
0
What should we do?
Reaction time (ms)
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Collect more data (larger sample size), which is really the only option at this point…
bin
200-210
210-220
221-230
231-240
241-250
251-260
261-270
271-280
281-290
291-300
301-310
311-320
321-329
330-339
sample size
non-athletes athletes
0
1
2
2
1
2
2
6
12
17
15
9
4
0
73
18
3
6
8
12
15
10
8
6
3
3
2
1
0
0
77
16
14
12
10
8
6
4
2
0
Series1
Nonathletes
Series2
Athletes
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Series2
Number of students (frequency)
Series1
200-210
210-220
221-230
231-240
241-250
251-260
261-270
271-280
281-290
291-300
301-310
311-320
321-329
330-339
Number of students (frequency)
18
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
16
14
12
10
8
Series1
6
Series2
4
2
0
Reaction time (ms)
Mean +/- σ
Reaction time (ms)
athletes
Nonathletes
264 +/- 30.6
279 +/- 31.6
Sample size: 25 in each group (N=50)
Mean +/- σ
athletes
Nonathletes
251 +/- 30.8
298 +/- 28.5
Sample size: 73 in non-athletes
77 in athletes
Descriptive
Histograms
and Statistics
Distributions
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
What you should notice is that the means changed dramatically and the two goups are beginning
to separate indicating that there may actually be a difference. There is no substitute for carefully
collected / high quality data and a large sample size.
Series2
Number of students (frequency)
Series1
200-210
210-220
221-230
231-240
241-250
251-260
261-270
271-280
281-290
291-300
301-310
311-320
321-329
330-339
Number of students (frequency)
18
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
16
14
12
10
8
Series1
6
Series2
4
2
0
Reaction time (ms)
Mean +/- σ
Reaction time (ms)
athletes
Nonathletes
264 +/- 30.6
279 +/- 31.6
Sample size: 25 in each group (N=50)
Mean +/- σ
athletes
Nonathletes
251 +/- 30.8
298 +/- 28.5
Sample size: 73 in non-athletes
77 in athletes
Descriptive Statistics
Measures of Spread
3. The STANDARD DEVIATION (σ or s)
Let’s go back to the small sample size data…
Number of students (frequency)
4.5
4
3.5
Mean +/- σ
athletes
Nonathletes
264 +/- 30.6
279 +/- 31.6
3
2.5
Series1
2
Series2
1.5
1
0.5
0
Reaction time (ms)
How can we determine if there is a significant difference between these two groups?
Histograms and Distributions
T-Test
assesses whether the means of two groups are statistically
different from each other
Histograms and Distributions
Histograms and Distributions
Histograms and Distributions
= Standard Error of the difference
Histograms and Distributions
Histograms and Distributions
Histograms and Distributions
Therefore the t-value is related to how different the means are and how
broad yours data is. A high t-value is obviously what you hope for…
Calculate the t-score
Histograms and Distributions
t = -1.61
-Degrees of freedom is the sum of the people in both groups minus 2
df = 48
Histograms and Distributions
The null hypothesis vs the hypothesis
1. The hypothesis:
Athletes will have a quicker reaction time than non-athletes.
2. The null hypothesis:
The null hypothesis always states that there is no
relationship between the two groups or there is no
difference in reaction time between athletes and nonathletes.
Histograms and Distributions
The p-value
1. The p-value is a number between 0 and 1.
2. It is the probability (hence the p-value) that there is no
difference between the groups supporting the null
hypothesis.
3. Therefore, the probability that there is a difference
between the two groups is 1 minus the p-value.
4. In order for the data to support the hypothesis, the
p-value must be high or low?
The p-value should be low (<0.05), which says that there is less than a 5% chance
that there is no difference between the two groups. Therefore, there is greater than
95% chance that there is a difference.
Histograms and Distributions
Statistical Significance
When the p-value is less than 0.05, we say that the data is
statistically significant, and there may be a real difference
between the two groups.
Be warned that just because p is less than 0.05 between two groups doesn’t
mean that there is actually a difference. For example, if we find p < 0.05 for the
reaction time experiment, it doesn’t mean that there is a definite difference
between athletes and non-athletes. It only means that there is a difference in our
data, but our data might be flawed or there is not enough data yet (sample size
too small) or we measured the data improperly, or the sampling wasn’t random,
or the experiment was garbage, etc…
Doubt is the greatest tool of any scientist (person).
Histograms and Distributions
How is the p-value determined?
The p-value is found by using a standard t-table in
combination with the t-value and the degrees of freedom
previously determined:
http://bioinfo-out.curie.fr/ittaca/documentation/Images/ttable.gif
http://davidmlane.com/hyperstat/t-table.html
http://www.graphpad.com/quickcalcs/Pvalue2.cfm
Histograms and Distributions
Now you determine the p-value for your data.
Histograms and Distributions
1. Begin by choosing the dependent variable like grade for
example.
Since the T-test can only look at two groups simultaneously and there are four grades,
we need to perform all the possible combinations (there was apparently only one 9th
grader and therefore the sample size is too low to look at this grade):
10th vs 11th
10th vs 12th
11th vs 12th
We also would want to know if the mean of each group is significantly different than the
actual value.
Actual value vs 10th
Actual value vs 11th
Actual value vs 12th
This needs to be done twice, once for the line estimation and once for the dots estimation!!
Histograms and Distributions
These are the tables you need to fill out:
Grade
Mean
SD
Variance
10th
11th
12th
Gades
Difference of
means
Variability
of Groups
T-score
P-value
10th vs actual
11th vs actual
12th vs actual
10th vs 11th
10th vs 12th
11th vs 12th
Write a conclusion based on your analysis. Remember, just because p < 0.5 it doesn’t
necessarily mean you hypothesis is supported!