page 58 of text Name the two values if the set is bimodal
Data skewed to the left is said to be ‘negatively skewed’ with the mean and median to the left of the mode. Data skewed to the right is said to be ‘positively skewed’ with the mean and media to the right of the mode.
Data not ‘lopsided’.
Data lopsided to left (or slants down to the left - definition of skew is ‘slanting’)
Data lopsided to the right (or slants down to the right)
Reminder: range is the highest score minus the lowest score
Reminder: range is the highest score minus the lowest score
Reminder: range is the highest score minus the lowest score
These ideas will be used repeatedly throughout the course.
page 79 of text
Some student have difficulty understand the idea of ‘within one standard deviation of the mean’. Emphasize that this means the interval from one standard deviation below the mean to one standard deviation above the mean.
These percentages will be verified by the concepts learned in Chapter 5. Emphasize the Empirical Rule is appropriate for data that is in a BELL-SHAPED distribution.
Transcript
1.
Single value in series of observations which indicate the characteristics of observations
All data / values clustered around it & used to compare between one series to another
Measures: a) Mean ( Arithmetic / Geometric / Harmonic)
If we have x 1 , x 2 , …… x n observations with corresponding frequencies f 1 , f 2 , ……f n, then
x 1 f 1 + x 2 f 2 + ……. x n f n Σfx
Mean = --------------------------------- = ----------
f 1 + f 2 + ……f n Σf
Problem : Calculate the avg. no. of children / family from the following data:
Mean = 519/ 240 = 2.163 No. of Children (X) No. of families ( f ) Total no of children (fx) 0 30 0 x 30 = 0 1 52 1 x 52 = 52 2 60 2 x 60 = 120 3 65 3 x 65 = 195 4 18 4 x 18 = 72 5 10 5 x 10 = 50 6 5 6 x 5 = 30 Total = 240 = 519
Data is symmetric if the left half of its histogram is roughly a mirror of its right half.
Skewed
Data is skewed if it is not symmetric and if it extends more to one side than the other.
Definitions
10.
Skewness Mode = Mean = Median SYMMETRIC Figure 2-13 (b)
11.
Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median Figure 2-13 (b) Figure 2-13 (a)
12.
Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median Figure 2-13 (b) Figure 2-13 (a) Figure 2-13 (c)
Range : difference between the highest and the lowest value
Problem:
Systolic and diastolic pressure of 10 medical students are as follows: 140/70, 120/88, 160/90, 140/80, 110/70, 90/60, 124/64, 100/62, 110/70 & 154/90. Find out the range of systolic and diastolic blood pressure
Solution:
Range of systolic blood pressure of medical students: 90-160 or 70
Range of diastolic blood pressure of medical students: 60-90 or 30
Mean Deviation: average deviations of observations from mean value
_
Σ (X – X ) __
Mean deviation (M.D) = --------------- , ( where X = observation, X = Mean
Calculate difference between each observation and mean
↓
Square the differences
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Sum the squared values
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Divide the sum of squares by the no. observations (n) to get ‘mean square deviation’ or variances (σ 2 ). [For sample size < 30, it will be divided by (n-1)]
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Find the square root of variance to get Root-Means-Square-Deviation or S.D ( σ)
20.
Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range 4 s or (minimum usual value) (maximum usual value)
21.
Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range 4 s or (minimum usual value) (maximum usual value) Range 4 s
22.
Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range 4 s or (minimum usual value) (maximum usual value) Range 4 s = highest value - lowest value 4
minimum ‘usual’ value (mean) - 2 (standard deviation)
minimum x - 2(s)
maximum ‘usual’ value (mean) + 2 (standard deviation)
maximum x + 2(s)
26.
x The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15
27.
x - s x x + s 68% within 1 standard deviation 34% 34% The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15
28.
x - 2s x - s x x + 2s x + s 68% within 1 standard deviation 34% 34% 95% within 2 standard deviations The Empirical Rule (applies to bell-shaped distributions ) 13.5% 13.5% FIGURE 2-15
29.
x - 3s x - 2s x - s x x + 2s x + 3s x + s 68% within 1 standard deviation 34% 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean The Empirical Rule (applies to bell-shaped distributions ) 0.1% 2.4% 2.4% 13.5% 13.5% FIGURE 2-15 0.1%
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