This document discusses computing statistics for single-variable data. It describes six common statistics: three measures of central tendency (mean, median, mode), two measures of spread (variance and standard deviation), and one measure of symmetry (skewness). Formulas are provided for calculating each statistic. Examples are given for computing statistics for both discrete and continuous data sets.

1. Computing Statistics
Quantitative Methods

2. Single-variable Statistics
 We will be considering six statistics of a data set
 Three measures of the middle
 Mean, median, and mode
 Two measures of spread
 Variance and standard deviation
 One measure of symmetry
 Skewness
 We can compute these values for either discrete or
continuous data.

3. Mean or Average
 The mean is defined as the sum of the data divided by the number of data
 The variable often used is m the Greek ‘mu’, or 𝑥 (X bar). Often m is
associated with a population and 𝑥 is associated with a sample.
 Symbolically, 𝑥 =
𝑥
𝑛
, where 𝑥 = 𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛, and n is the number
of data values. (The capital letter sigma,S ,represents summation.)
 Example: Data is (1, 2, 3, 4, 5). The sum is 1+2+3+4+5=15. There are 5
data values, so the average is 15/5=3.
 note= Many calculators have a ‘statistics’ mode. The way the manufacturer
chooses to implement statistical calculation varies widely.

4. Median
 The median is the middle number when the data is listed in order. If there is an even
number of data points, the median is the average of the two middle values.
 Example: Data is (1,2,3,4,5). Odd numbers - The median is 3
 Example: Data is (1,2,3,4,5,6). Even numbers -The median is (3+4)/2=3.5
 Why is this quantity useful?
 The median ignores outlying values. What if our data had been (1,2,3,4,1000)?
 The mean is 202, which is not characteristic of any of the actual values.
 The median is 3, which is more typical of most of the values.
 The median is helpful when looking for a house to buy. The median house price is the
typical price you’d pay, even though the millionaire’s house at the corner of the block
raises the mean of the house prices above the value most people paid for theirs.

5. Mode
 The mode represents the most populated class, or the group with the most members. This is yet
another reasonable way of finding the middle of the data.
 Determining the mode is different for discrete data than it is for continuous data.
 For discrete data, the mode is simply the number that appears the most times.
 Data is (1, 1, 2, 3, 4, 4, 5, 5, 5). The mode is 5.
 For continuous data, the mode is the center of the range of the class that has the most members in
it.
 Data is (1.1, 1.2, 1.3, 1.8, 2.0, 2.6, 3.1, 4.6, 4.8, 5.1). The class from 1-2
has the most members. The center of this range is 1.5, so the mode is 1.5. (Note: 1.5 does not
even appear in the data.)
 In both cases, the mode can be quickly determined from the graph. The mode is the x-value that is at
the center of the tallest bar in either the bar graph (discrete data) or histogram (continuous data).
 Data can have two modes (bi-modal), but if there are more, we usually say it is amodal (no distinct
mode).
0
1
2
3
4
1 2 3 4 5

6. Variance
 Variance (is the power of 2 standard deviation) (var. or s2 or s2) is a measure of the spread
of data about the average. We don’t care which direction the difference is, so we will be
ignoring the sign of the difference. In words, the variance is the sum of the squares of the
differences divided by one less than the number of data values.
 The equation is 𝑣𝑎𝑟. =
(𝑥−𝑥)2
𝑛−1
𝑥 𝑥 𝑥 − 𝑥 (𝑥 − 𝑥)2
1
2
3
4
5
• Example: Data is (1, 2, 3, 4, 5) and
mean (𝑥 =xbar) is 3.
• Variance is 10/(5-1)=2.5
• If you are using a calculator, it is most likely
that the calculator will compute the standard
deviation (s) instead. To get the variance
from the standard deviation, simply find the
square of the standard deviation:
• 𝑣𝑎𝑟 = 𝜎2
(var is the power of 2 of st dev)
𝑥 𝑥 𝑥 − 𝑥 (𝑥 − 𝑥)2
1 3
2 3
3 3
4 3
5 3
𝑥 𝑥 𝑥 − 𝑥 (𝑥 − 𝑥)2
1 3 -2
2 3 -1
3 3 0
4 3 1
5 3 2
𝑥 𝑥 𝑥 − 𝑥 (𝑥 − 𝑥)2
1 3 -2 4
2 3 -1 1
3 3 0 0
4 3 1 1
5 3 2 4
𝑥 𝑥
xbar
𝑥 − 𝑥 (𝑥 − 𝑥)2
1 3 -2 4
2 3 -1 1
3 3 0 0
4 3 1 1
5 3 2 4
10

7. Standard Deviation
 Standard deviation (std. dev. or s or s) is a measure of the spread of data about the
average. We don’t care which direction the difference is, so we will be ignoring the sign of
the difference. In words, the standard deviation is the square root of (the sum of the
squares of the differences divided by one less than the number of data values).
 The equation is 𝑠𝑡𝑑. 𝑑𝑒𝑣. =
(𝑥−𝑥)2
𝑛−1
= 𝑣𝑎𝑟. (st dev is the rooth square of variance)
 Example (from previous slide): Data is (1, 2, 3, 4, 5), mean (𝑥) is 3, and we previously
found that the variance is 𝑣𝑎𝑟. =2.5
 Since the standard deviation is the square root of variance,
 Standard deviation is σ = 2.5 = 1.58
 If you are using a calculator, it is most likely that the calculator will compute the standard
deviation (s) as part of its normal statistical function. There is a tutorial for using this
course’s standard calculator, the TI-30Xa, to calculate standard deviation.
 Question: Since standard deviation and variance differ by one keystroke or key button , why
do we need both?
 The units of standard deviation are the same as the data. Variance has other direct uses
(e.g. Analysis of Variance) and is also more easily computed.

8. Skewness
 The distribution of a set of data may have symmetry about the mean, or it may have a longer
‘tail’ to one side or the other.
 Imagine draping a sheet over the graph of the data. The side of the sheet that is least steep
is the side that has the longer tail.
 If the tail points to the right (toward positive x values), the skewness will be a positive
number.
 If the tail points to the left, skewness will be negative.
 Zero skewness indicates symmetric tails to both sides.
 It is sometimes difficult to estimate from the graph what the skewness will be, but there is a
formula for calculating skewness in all cases:
 Skewness = (mean-mode)/(standard deviation)
Data is (1.1, 1.2, 1.3, 1.8, 2.0,
2.6, 3.1, 4.6, 4.8, 5.1).
Mean is 2.76 (summation /n)
Mode is 1.5 (most repetted)
Std. Dev. is 1.56 = 𝑣𝑎𝑟.
Skewness =
(2.76−1.5)
1.56
= 0.81
(tail to the right)

9. Example: Discrete Data
 Data: 1, 1, 2, 3, 3, 4, 4, 4, 5
 N:
 Graph:
 Mean:
 Median:
 Mode:
 Variance:
 Standard Deviation:
 Skewness:
0
1
2
3
4
1 2 3 4 5
𝑥 𝑥 𝑥 − 𝑥 (𝑥 − 𝑥)2
1 3 -2 4
1 3 -2 4
2 3 -1 1
3 3 0 0
3 3 0 0
4 3 1 1
4 3 1 1
4 3 1 1
5 3 2 4
27 16

10. Example: Discrete Data
 Data: 1, 1, 2, 3, 3, 4, 4, 4, 5
 N: 9
 Graph:
 Mean: 3
 Median: 3
 Mode: 4
 Variance: 2
 Standard Deviation: 1.41
 Skewness: -0.71
0
1
2
3
4
1 2 3 4 5
𝑥 𝑥 𝑥 − 𝑥 (𝑥 − 𝑥)2
1 3 -2 4
1 3 -2 4
2 3 -1 1
3 3 0 0
3 3 0 0
4 3 1 1
4 3 1 1
4 3 1 1
5 3 2 4
27 16

11. Example: Continuous Data
 Data: 1.5, 1.7, 2.4, 2.5, 2.7, 3.5, 3.8, 4.7, 5.1, 5.1
 N:
 Graph:
 Mean:
 Median:
 Mode:
 Variance:
 Standard Deviation:
 Skewness:
𝑥 𝑥 𝑥 − 𝑥 (𝑥 − 𝑥)2
1.5 3.3 -1.8 3.24
1.7 3.3 -1.6 2.56
2.4 3.3 -0.9 0.81
2.5 3.3 -0.8 0.64
2.7 3.3 -0.6 0.36
3.5 3.3 0.2 0.04
3.8 3.3 0.5 0.25
4.7 3.3 1.4 1.96
5.1 3.3 1.8 3.24
5.1 3.3 1.8 3.24
Skewness = (mean-mode)/(stand dev)
Mean is (summation /n)
Mode is (most repeated)
𝑠𝑡𝑑. 𝑑𝑒𝑣. =
(𝑥 − 𝑥)2
𝑛 − 1
= 𝑣𝑎𝑟.
𝑥 =
𝑥
𝑛

12. Example: Continuous Data
 Data: 1.5, 1.7, 2.4, 2.5, 2.7, 3.5, 3.8, 4.7, 5.1, 5.1
 N: 10
 Graph:
 Mean: 3.3
 Median: 3.1
 Mode: 2.5
 Variance: 1.81
 Standard Deviation: 1.35
 Skewness: 0.6
𝑥 𝑥 𝑥 − 𝑥 (𝑥 − 𝑥)2
1.5 3.3 -1.8 3.24
1.7 3.3 -1.6 2.56
2.4 3.3 -0.9 0.81
2.5 3.3 -0.8 0.64
2.7 3.3 -0.6 0.36
3.5 3.3 0.2 0.04
3.8 3.3 0.5 0.25
4.7 3.3 1.4 1.96
5.1 3.3 1.8 3.24
5.1 3.3 1.8 3.24
Skewness = (mean-mode)/(stand dev)
Mean is (summation /n)
Mode is (most repeated)
𝑠𝑡𝑑. 𝑑𝑒𝑣. =
(𝑥 − 𝑥)2
𝑛 − 1
= 𝑣𝑎𝑟.
𝑥 =
𝑥
𝑛

13. Descriptive statistics for selected variables data
MVA PCI RT VAA
Mean 2.048343 1.732004 0.006779 4.265940
Median 2.511274 1.921934 0.010922 3.408486
Maximum 6.709957 4.575449 0.083430 14.05086
Minimum -10.82965 -4.261222 -0.063971 5.469763
Std. Dev. 3.309870 1.878032 0.030488 5.072581
Skewness -1.489831 -0.883407 -0.143132 0.183532
Kurtosis 6.650875 3.815743 2.716093 2.382058
tween 1970 and 2014 (mio)

14. How to interpret descriptive stats
 When the descriptive statistics in Table are
considered,
 mean and median values both measure the
central tendency of the considered data to
compare and determine which data is the
better measure to use

15. How to interpret Descriptive Stats
 If the selected data are symmetric, then the
mean and median values are expected to be
similar.
 In this study for PCI value, the median for
manufacturing value added (MVA) and for retail
trade (RT) is higher than the mean, which implies
that the skew is to the left,
 the mean is left of the median and it is lower.

16. How to interpret descriptive stats
 these parameters are assumed to be asymmetric
with a long tail on the left because the estimated
median value is greater than the mean and
negative skewness is observed.

17. How to interpret descriptive stats
 The small difference between standard deviation
and media is good because it supports the success
of estimation.
 However large difference is the indication of
heterogeneity among residuals and it is not good.

18. How to interpret descriptive stats
 It is also known that the mode, median, and mean
do not coincide in skewed distributions, although
their relative positions remain constant - moving
away from the `peak’ and toward the `tail,’ the
order is always from mode, to median, to mean

19. Thank you
eakalpler@gmail.com
erginakalpler@csu.edu.tr

20. Conclusion
 We can answer a great deal of statistical questions by examining the graph and six standard
statistical variables for the data:
 Bar graph or histogram
 Measures of the middle
 Mean (can be done on a calculator)
 Median (obtained from the sorted list of data)
 Mode (obtained from the graph)
 Measures of the spread
 Variance (calculated using a tabular method) [or the square of the std. dev.]
 Standard Deviation (obtained from calculator’s statistics mode) [or the square root of
the variance]
 Measure of symmetry
 Skewness (calculated from the above values Mean, Mode, and Std. Dev.)