Standard

Deviation,

Standard

Error

Which

'Standard'

Should

We

Use?

George

W.

Brown,

MD

\s=b\

Standard

deviation

(SD)

and

standard

error

(SE)

are

quietly

but

extensively

used

in

biomedical

publications.

These

terms

and

notations

are

used

as

descriptive

sta-

tistics

(summarizing

numerical

data),

and

they

are

used

as

inferential

statistics

(esti-

mating

population

parameters

from

sam-

ples).

I

review

the

use

and

misuse

of

SD

and

SE

in

several

authoritative

medical

journals

and

make

suggestions

to

help

clarify

the

usage

and

meaning

of

SD

and

SE

in

biomedical

reports.

(Am

J

Dis

Child

1982;136:937-941)

Standard

deviation

(SD)

and

stan¬

dard

error

(SE)

have

surface

simi¬

larities;

yet,

they

are

conceptually

so

different

that

we

must

wonder

why

they

are

used

almost

interchangeably

in

the

medical

literature.

Both

are

usually

preceded

by

a

plus-minus

symbol

(±),

suggesting

that

they

define

a

sym¬

metric

interval

or

range

of

some

sort.

They

both

appear

almost

always

with

a

mean

(average)

of

a

set

of

measure¬

ments

or

counts

of

something.

The

med¬

ical

literature

is

replete

with

statements

like,

"The

serum

cholesterol

measure¬

ments

were

distributed

with

a

mean

of

180±30

mg/dL

(SD)."

In

the

same

journal,

perhaps

in

the

same

article,

a

different

statement

may

appear:

"The

weight

gains

of

the

sub¬

jects

averaged

720

(mean)

±32

g/mo

(SE)."

Sometimes,

as

discussed

further,

the

summary

data

are

presented

as

the

"mean

of

120

mg/dL

±12"

without

the

"12"

being

defined

as

SD

or

SE,

or

as

some

other

index

of

dispersion.

Eisen¬

hart1

warned

against

this

"peril

of

From

the

Los

Lunas

Hospital

and

Training

School,

New

Mexico,

and

the

Department

of

Pedi-

atrics,

University

of

New

Mexico

School

of

Medi-

cine,

Albuquerque.

Reprint

requests

to

Los

Lunas

Hospital

and

Training

School,

Box

1269,

Los

Lunas,

NM

87031

(Dr

Brown).

shorthand

expression"

in

1968;

Fein-

stein2

later

again

warned

about

the

fatuity

and

confusion

contained

in

any

a

±

b

statements

where

b

is

not

defined.

Warnings

notwithstanding,

a

glance

through

almost

any

medical

journal

will

show

examples

of

this

usage.

Medical

journals

seldom

state

why

SD

or

SE

is

selected

to

summarize

data

in

a

given

report.

A

search

of

the

three

major

pediatrie

journals

for

1981

(Amer¬

ican

Journal

of

Diseases

of

Children,

Journal

of

Pediatrics,

and

Pediatrics)

failed

to

turn

up

a

single

article

in

which

the

selection

of

SD

or

SE

was

explained.

There

seems

to

be

no

uniformity

in

the

use

of

SD

or

SE

in

these

journals

or

in

The

Journal

of

the

American

Medical

Association

(JAMA),

the

New

England

Journal

of

Medicine,

or

Science.

The

use

of

SD

and

SE

in

the

journals

will

be

discussed

further.

If

these

respected,

well-edited

jour¬

nals

do

not

demand

consistent

use

of

either

SD

or

SE,

are

there

really

any

important

differences

between

them?

Yes,

they

are

remarkably

different,

despite

their

superficial

similarities.

They

are

so

different

in

fact

that

some

authorities

have

recommended

that

SE

should

rarely

or

never

be

used

to

sum¬

marize

medical

research

data.

Fein-

stein2

noted

the

following:

A

standard

error

has

nothing

to

do

with

standards,

with

errors,

or

with

the

commu¬

nication

of

scientific

data.

The

concept

is

an

abstract

idea,

spawned

by

the

imaginary

world

of

statistical

inference

and

pertinent

only

when

certain

operations

of

that

imagi¬

nary

world

are

met

in

scientific

reality.2(p336)

Glantz3

also

has

made

the

following

rec¬

ommendation:

Most

medical

investigators

summarize

their

data

with

the

standard

error

because

it

is

always

smaller

than

the

standard

deviation.

It

makes

their

data

look

better

.

.

.

data

should

never

be

summarized

with

the

stan¬

dard

error

of

the

mean.3*"25™

A

closer

look

at

the

source

and

mean¬

ing

of

SD

and

SE

may

clarify

why

medical

investigators,

journal

review¬

ers,

and

editors

should

scrutinize

their

usage

with

considerable

care.

DISPERSION

An

essential

function

of

"descriptive

statistics"

is

the

presentation

of

con¬

densed,

shorthand

symbols

that

epito¬

mize

the

important

features

of

a

collec¬

tion

of

data.

The

idea

of

a

central

value

is

intuitively

satisfactory

to

anyone

who

needs

to

summarize

a

group

of

measure¬

ments,

or

counts.

The

traditional

indica¬

tors

of

a

central

tendency

are

the

mode

(the

most

frequent

value),

the

median

(the

value

midway

between

the

lowest

and

the

highest

value),

and

the

mean

(the

average).

Each

has

its

special

uses,

but

the

mean

has

great

convenience

and

flexibility

for

many

purposes.

The

dispersion

of

a

collection

of

values

can

be

shown

in

several

ways;

some

are

simple

and

concise,

and

others

are

com¬

plex

and

esoteric.

The

range

is

a

simple,

direct

way

to

indicate

the

spread

of

a

collection

of

values,

but

it

does

not

tell

how

the

values

are

distributed.

Knowl¬

edge

of

the

mean

adds

considerably

to

the

information

carried

by

the

range.

Another

index

of

dispersion

is

pro¬

vided

by

the

differences

(deviations)

of

each

value

from

the

mean

of

the

values.

The

trouble

with

this

approach

is

that

some

deviations

will

be

positive,

and

some

will

be

negative,

and

their

sum

will

be

zero.

We

could

ignore

the

sign

of

each

deviation,

ie,

use

the

"absolute

mean

deviation,"

but

mathematicians

tell

us

that

working

with

absolute

num¬

bers

is

extremely

difficult

and

fraught

with

technical

disadvantages.

A

neglected

method

for

summarizing

the

dispersion

of

data

is

the

calculation

of

percentiles

(or

deciles,

or

quartiles).

Percentiles

are

used

more

frequently

in

pediatrics

than

in

other

branches

of

medicine,

usually

in

growth

charts

or

in

other

data

arrays

that

are

clearly

not

symmetric

or

bell

shaped.

In

the

gen¬

eral

medical

literature,

percentiles

are

sparsely

used,

apparently

because

of

a

common,

but

erroneous,

assumption

that

the

mean

±

SD

or

SE

is

satisfactory

for

summarizing

central

tendency

and

dispersion

of

all

sorts

of

data.