This document discusses statistical concepts like frequency, mean, standard deviation, normal distribution, z-scores, and outliers. It provides the descriptive statistics of a survey measuring job satisfaction on a 5-point scale. The mean was 4.048, with most responses between 3-5. However, some outliers with extremely low values of 1.17 were identified, questioning the normality of the distribution. Various graphs and tests were used to further analyze the data distribution and identify outliers.

1. Frequencies

2. The number of occurrence of an event per unit
of time.
Means how many people responding same
answer during data collection. For example if
someone collecting data about gender, the
number of male/female is the frequency of M
and F in this survey.

3. Statistics
Job satisfaction
N Valid 264
Missing 14
Mean 4.0480
Std. Error of Mean .03883
Median 4.1667
Mode 4.00
Std. Deviation .63086
Variance .398
Skewness -1.592
Std. Error of Skewness .150
Kurtosis 4.224
Std. Error of Kurtosis .299
Range 3.83
Minimum 1.17
Maximum 5.00
Sum 1068.67
Percentiles 25 3.8333
50 4.1667
75 4.5000

4. Analyze > Descriptive Statistics > Frequencies

6. Std Error of Means
Standard deviation of sample means taken from
the population is called “Std Error of Means”.
The smaller the std error the more sample will
represent the population.
Std Error = Std Deviation/Sqr root of the size of
the population = σ/√N
Where σ = 𝑥𝑖−µ
2
𝑁

7. Mean and Standard Deviation Role in Normal
Distribution:
Normal Distribution Require Bell shape
distribution of data, which is possible only
Pr µ − 1σ ≤ 𝑋 ≤ µ + 1σ ≈ 0.6827
Or µ±1 σ
Pr µ − 2σ ≤ 𝑋 ≤ µ + 2σ ≈ 0.9545
Or µ±2 σ
Pr µ − 3σ ≤ 𝑋 ≤ µ + 3σ ≈ 0.9973
Or µ±3 σ
The above is also called rule of 68-95-99.7

8. Putting the value of Mean and Standard
Deviation, we conclude that
68% of data falls between 3.41714 and
4.67886.
95% of data falls between 2.78628 and
5.30972.
99.7% of data falls between
2.1554200000000003 and 5.94058.

10. Z-Score
Z-score tell us how much standard deviation
actual value far way from mean.
Formula
Z = (X-µ)/σ for Population
z=
𝑥−𝑥′
𝑆
for sample
Example:
x = 6
µ = 4.0480, σ = .63086

11. z = (6-4.048)/.631
Z= 3.09
Interpretation
The value of that particular variable is about 3
standard deviation above the mean
Find an area under a normal curve from z=0
to z=?
This can be done from z-table:
Z=0 means that actual value is equal to mean
value

12. Z calculated value is 3.09, so from table we can
calculate the percentage for area under the
normal curve
z table
Means that
49.9% of the
Area on the
Right side fall
Under curve
z 0.00 0.01 0.02 .03 - - - 0.
08
0.09
0.0 ↓
0.1 ↓
0.2 ↓
0.3 ↓
- ↓
1.0 ↓
1.1 ↓
- ↓
2.0 ↓
2.1 ↓
- ↓
3→ → → → → → → → → 0.499

13. If we want to check for both side
=.499+.5=0.999 or 99.9% of area fall under
normal curve on both side.

14. Outliers
In statistics, an outlier is a data point that differs
significantly from other observations.
It is because of either
(i) Variability in the measurement
(ii) Experimental error

15. Detection
There are various methods to detect outliers,
some are graphical like normal probability plots
and some are model based like Box Plots.
Analyze >> Descriptive >> Explore

19. Result

20. Interpretation:
The mean value of the employees’ responses on
job satisfaction averages at 4.048; the value falls
between 4 (I Agree) and 5 (I strongly Agree). The
values of Skewness (S) and Kurtosis (K),
respectively are -1.592 and 4.224, while a
normal distribution requires these values to be
equal to 0 and 3.

22. EXTREME VALUES:
The highest extreme values in this case are
logically acceptable, but the value of
observation No. 31 and 229 are extremely low,
each one is equal to 1.17; a third observation
No.228 also has a low value (1.50).

24. Interpretation:
Out of the two tests, the latter test (Shapiro-
Wilk Test) is considered more appropriate for
small sample sizes (< 50 samples) but it can also
handle sample sizes as large as 2000.
In both test cases, if the Sig. value of the test is
less than 0.05, then we reject null hypothesis
“the data is not normal” and accept alternative
“the data is normal”. If it is greater 0.05, then
the data significantly deviate from a normal
distribution.

26. Interpretation:
The Histogram reflects that most of the
responses lie within the values of 3 and 5, with
the exception of a few which appear lying on
extreme left side, between values of 1 and 2.

28. Interpretation:
This plot of Stem and Leaf shows that there are
some extreme cases especially on lower side,
suggesting that 16% responses came with the
value of below 3, and remaining 84% lies
between 3 and 5.

30. Interpretation:
From this graph we can conclude that the data
mostly appear to be normally distributed as it
follows the diagonal line with the exception of
some portions where data appear away from
the straight diagonal line.
The detrended Normal Q-Q Plot, provided on
next slide, further clarifies the position.

32. Interpretation:
In previous slide we see that a very little
observation fall above the trend line, and most
of the observations are under trend line,
showing the normality characteristic of the
observations.

34. Interpretation:
The box plot on previous slide distinguishes
between majority of the cases which lied
between values of 3 to 5, and those fell below 3;
this plot helps identify all the cases having
values below 3, as well as, the three cases
having values below 2.