What is Central Tendency?

Why needed?

##                    mpg cyl  disp  hp drat
## Mazda RX4         21.0   6 160.0 110 3.90
## Mazda RX4 Wag     21.0   6 160.0 110 3.90
## Datsun 710        22.8   4 108.0  93 3.85
## Hornet 4 Drive    21.4   6 258.0 110 3.08
## Hornet Sportabout 18.7   8 360.0 175 3.15
## Valiant           18.1   6 225.0 105 2.76
## Duster 360        14.3   8 360.0 245 3.21
## Merc 240D         24.4   4 146.7  62 3.69

##       mpg             cyl           disp      
##  Min.   :14.30   Min.   :4.0   Min.   :108.0  
##  1st Qu.:18.55   1st Qu.:5.5   1st Qu.:156.7  
##  Median :21.00   Median :6.0   Median :192.5  
##  Mean   :20.21   Mean   :6.0   Mean   :222.2  
##  3rd Qu.:21.75   3rd Qu.:6.5   3rd Qu.:283.5  
##  Max.   :24.40   Max.   :8.0   Max.   :360.0

Criteria for a Good Measure of Central Tendency

• Well-defined

• Understandable
• Considers all values
• Suitable for further analysis
• Not affected by sample fluctuation

Measures (Averages)

• Arithmetic Mean (AM)
• Geometric Mean (GM)
• Harmonic Mean (HM)
• Median
• Mode
• Partition Values (Quartiles, Deciles, Percentiles etc.)

AM

\(AM=\bar x=\frac{\sum x}{n}\)

If there are frequencies or weights

\(\bar x=\frac{\sum f_i x_i}{\sum f_i} \space or \space \frac{\sum w_i x_i}{w_i=n}\)

Find AM

##  [1] 10 55 38 48 51 25 14 44 23 22

## [1] 33

Find AM: 2, 2, 3, 3, 5, 5, 5, 8, 8, 9

There are 2 ways.

## [1] 5

Mean Using Frequency

For grouped data

\(\therefore \bar x = \frac{\sum fx}{\sum f} =\frac{50}{10}=5\)

Freuency vs Weight

## [1] 5.93

Shortcut Method for AM

## [1] 1009 1037 1047 1024 1013 1043

## [1] "The new values are"

## [1] -11  17  27   4  -7  23

## [1] "Mean of y is 8.83"

## [1] "Mean of x is 1028.83"

Shortcut Method Formula

Properties of AM

Also look at this

• \(\sum (x_i-\bar x)=0\); can you prove it?

• \(\sum (x_i-\bar x)^2 \le \sum (x_i-a)^2, \space a \ne \bar x\)
• Depends on change of origin and scale?
• \(\bar x + \bar y =\frac{\sum x+\sum y}{n_x+n_y}\)
• Combined mean: \(\bar x_c=\frac{n_1 \bar x_1+n_2 \bar x_2+...+n_k \bar x_k}{n_1+n_2+...+n_k}\)
• \(AM\ge GM \ge HM\) & \(AM \times HM = (GM)^2\)
• AM of first n natural numbers = \(\frac{n+1}{2}\)

Arithmetic Mean of Series

\(11,13,15, \cdots, 57\)

Tip: Let \(c=2nd \space Term - 1st \space Term\) and \(a=1st \space Term -c\)

\(U = \frac{X-a}{c}\)

(Dis)advantages of AM

• Well-defined
• Less affected by sample fluctuation
• Comparison among sets is easy
• Uses all values
• Suitable for further analysis.
• Affected by outliers

Concept of Logarithm

• Log and Antilog
• \(log_24=?\)
• if \(log_2x=3\), then x=?

GM using Log

• \(GM=(x_1 \times x_2 \times \cdots \times x_n)^{(1/n)}\)
• \(log (GM) = \frac 1n \{log(x_1)+log(x_2) + \cdots + log(x_n)\} = A (suppose)\)
• \(\therefore GM = Antilog (A)\)

GM Easier Formula

x = 20020, 30080, 50086, 40130

• log x = 9.9, 10.31, 10.82, 10.6
• Mean of logx = \(\frac{\sum logx}{n}\)
• Original Mean = \(antilog(\frac{\sum logx}{n})\)

Calculate GM

(Dis)advantages of GM

• Not affected by outliers

  x = 5, 10, 15, 20, 100, 1000

  log(x) = 0.7, 1, 1.18, 1.3, 2, 3

• Less affected by sample fluctuation

• Suitable for further analysis

• What if one or some x = 0?

• What if one or some x < 0?

Velocity Paradox

Story of oil scam

Harmonic Mean

Why and When HM

• When there are rates associated, say speed, and numerator is fixed.
• \(Speed, v = \frac{S}{t}\); HM if S is fixed
• Example: A man travels 120 km the first day at 12 kph, the same distance at 10 kph on the 2nd day, and at 8 kph on the 3rd day. Find his average speed.

Wighted AM vs Weighted HM

Suppose, a bus travels 10 km at 10 kph, another 15 km at 20 kph, and another 20 km at 25 kph. What is the average speed.

HM \(\rightarrow\) consider distances as weights

AM \(\rightarrow\) consider times as weights

Time, \(t=\frac d v=\) 1, 0.75, 0.8

• \(WHM = \frac{10+15+20}{\frac{10}{10}+\frac{15}{20}+\frac{20}{25}}\)=17.65
• \(WAM = \frac{1\times 10 + 0.75 \times 20 + 0.80 \times 25}{1+0.75+0.80}\) = 17.65
• True mean, \(\bar v=\frac{\sum d}{\sum t}=\frac{45}{2.55}=\) 17.65

HM Example 2

• A passerby travels 10 km at 20 kph, 5 km at 15 kph, and 4 km at 12 kph. What is the average speed? (Use weighted HM)

• Here, distances are different. Consider them weights.

• \(HM=\frac{w_1+w_2+w_3}{\frac{w_1}{x_1}+\frac{w_2}{x_2}+\frac{w_3}{x_3}}\)

• HM = 16.286

Quadratic Mean

\(QM=\sqrt{\frac{x_1^2+x_2^2+...+x_n^2}{n}}=\sqrt{\frac{\sum x_i^2}{n}}\)

• Also known as Root mean square
• More at: https://en.wikipedia.org/wiki/Root_mean_square#Uses
• For grouped data?

Find medians

x = 4,5,6,8,9,11,16

y = 4,5,6,8,9,11,16,19

• Median of x is 8
• Median of y is 8.5
• Formula for odd n = \(\frac{n+1}{2}th \space value\)
• Formula for even n = \(\frac{\frac{n}{2}th+(\frac{n}{2}+1)th}{2}\)

Dis(advantages) of Median

• Unaffected by outliers (extreme values)
• Graphically estimable
• Affected by sample fluctuation
• Not based on all values
• Not suitable for further mathematical analysis

Quartiles, Deciles, and Percentiles

\(Q_1= \frac{(n+1)}{4}th \space\)

\(Q_2, Q_3=?\)

Find \(Q_1, Q_2, Q_3\)

X = 4, 6, 7, 10, 12, 13, 14, 15, 16, 17, 19

See more partition names

• 4, 8.5, 13, 12.0909091, 15.5, 19
• What are the formulae for Deciles (\(D_i\) and \(P_i\))

General Formula for Partition Values

where, k = no. of partitions

For median, for example, k = 2.

• What if we divide the data set into 20 segments?

Find Deciles and Percentiles

X = 4, 6, 7, 10, 12, 13, 14, 15, 16, 17, 19

Find \(D_3, D_8, P_{17}, P_{56}, P_{93}\)

Averages from Grouped Data

Partition Values from Graph

• Draw an Ogive
• Draw a straight line parallel to X-axis at the point \(\frac n 2\) of Y-axis. - Draw a perpendicular on X-axis from the intersection point.

When AM = GM = HM

If \(x_1=x_2=x_3\); prove it

Theorem 01

\[\sum_{i=0}^n (x_i-\bar x)=0\]

Theorem 02

\[\sum_{i=0}^n f_i(x_i-\bar x)=0\] ## Theorem 03

\[\sum_{i=1}^n (x_i-\bar x)^2 \lt \sum_{i=1}^n (x_i-a)^2; a\ne\bar x\]

\[\sum_{i=1}^n f_i(x_i-\bar x)^2 \lt \sum_{i=1}^n f_i(x_i-a)^2; a\ne\bar x\]

Theorem 04 (AM~origin & scale)

Prove that AM depends on origin and scale Use frequency as well i.e,

• \[\bar x=\frac {\sum_{i=1}^nx_i}{n}\]
• \[\bar x=\frac {\sum_{i=1}^n f_ix_i}{n}\]

Theorem 07

If the GM of \(n_1\) va;ues is \(G_1\), and of \(n_2\) values is \(G_2\), show GM of \(n_1+n_2\) values is \(G=\sqrt{G_1G_2}\)

Theorem 08

Theorem 09

For two non-zero positive numbers, \(AM \times HM =(GM)^2\)

Theorem 10

Mean and Median of first n natural numbers are \(\frac {n+1} 2\)

Theorem 11

If \(\bar x_1\) and \(\bar x_2\) are means of 2 data sets of sizes \(n_1\) and \(n_2\), respectively, the combined mean is \(\bar x_c=\frac{n_1 \bar x_1+n_2 \bar x_2}{n_1+n_2}\)

Theorem 12

If \(u=x+y, \bar u=\bar x + \bar y\); if \(n_1=n_2=n\)

Given \(u=x+y\)

\[\begin{eqnarray} \bar u &=& \frac{\sum u}{n} \nonumber \\ &=& \frac{\sum (x+y)}{n} \nonumber \\ &=& \frac{\sum x}{n}+ \frac{\sum y}{n} \nonumber \\ &=& \bar x + \bar y \nonumber \\ \end{eqnarray}\]

Theorem 13

For equal number of observations, GM of two variables is equal to the product of their individual means.

Theorem 14

\(GM=Antilog(\frac{\sum \log x_i}{n})\) or \(Antilog(\frac{\sum f_i \log x_i}{\sum f_i})\)

Theorem 15

If \(y = a + bx, \bar y = a + b \bar x\)

Theorem 16

If \(z_i=ax_i+by_i, \bar z=a \bar x + b \bar y\)

Example Problem 01

Example Problem 15

The mean of 200 numbers was 50. Later it was revealed that two observations were incorrectly given as 92 and 8, instead of 192 and 88, respectively. Find the correct mean.

Example Problem 22

If \(\sum f_i(x_i-k)=0\), what is the value of k?